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(     A 


FIELD-MANUAL 

FOR 

KAILROAD   ENGINEERS, 


BY 

J.  C.  NAGLE,  M.A.,  M.C.E., 

professor  nf  Civil  Knaineenna  if  the  Agricultural 
and  Medianicat  Culleye  of  Texas. 


SECOXD    KlUTIOX,   HE  VISED. 
SECOXI)    T1JOLSAJ5-D. 


KEW  YORK: 

JOHN   AVILEY   &   SONS. 

London:    CHAPMAN  &  HALL,   Limited. 

1903. 


Copyright,  1897 

BY 

J.   C.   NAGLE. 


ROBERT   DRUMMOND,    ELECTROTYPER   AND   PRINTER,    NEW   YGnK. 


PREFACE. 


Ease  of  reference  and  uniformity  of  notation  are  essential  in  a 

book  that  is  to  be  consulted  in  the  field.     With  this  in  mind  an 

effort  has  been  made  in  the  following  pages  to  secure  a  systematic 

arrangement  of  the  subject-matter  and  uniformity  of  terms  and 

-  notation.     Except  for  a  few  cases  Greek  letters  have  been  avoided 

"'  and  a  single  letter  is  used  to  designate  an  angle.     In  so  far  as 

"j)racticable  each  figure  is  intended  to  be  self-explanatory,  so  that 

^the  explanations  necessary  in  connection  with  the  problems  have 

->  been  reduced  to  a  minimum.     Algebraic  equations  stand  each  in 

ra  distinct  line,  thus  rendering  them  more  easily  read. 

t     A  knowledge  of  the  elements  of  geometry  and  trigonometry  has 

^been  assumed,  and  only  in  the  derivation  of  a  few  formulas  in 

^^connection  with  the  theory  of  transition-curves  will  any  higher 

mathematics  be  needed.     But  these  formulas  may  be  accepted  by 

the  reader  who  is  unfamiliar  with  the  calculus  without  in  any 

way  affecting   his  ability  to  understand  their  applications  or  to 

follow  sabst>quent  reasoning. 

rj     One  can  most  readily  turn  to  what  he  wants  in  a  book  after  hav- 

<  ing  become  familiar  with  its  contents  in  the  classroom.     Keeping 

"^  this  in  mind  this  book  has  been  written  so  that  it  may  be  used  as 

a  text  as  well  as  for  reference  in  the  field.     Wherever  practicable 

-^.solutions  to  problems  have  been  given  in  a  rigid,  general  form, 

5'  followed  by  illustrative  examples,  so  that  the  student  need  not 

viose  sight  of  the  principle  involved  while  following  the  solution 

£  for  a  particular  case.     Wherever  approximate  solutions  seemed 

■  preferable  they  have  also  been  given  and  their  limitations  pointed 

'^ut. 

Free  use  has  been  made  of  the  Table  of  Functions  of  a  One- 
degree  Curve,  thus  reducing  the  labor  of  field  computations.  By 
defining  the  degree  of  curve  with  reference  to  short  chords  for 

I  I  45577  '" 


IV  PREFACE^ 

sharp  curves — and,  with  tables  of  Radii,  Long  Chords,  Mid- 
ordinates,  etc.,  based  on  appropriate  equations — the  errors  result- 
ing from  assuming  the  radius  to  vary  inversely  with  the  degree 
of  curve  will  generally  be  found  to  be  quite  small. 

Chapter  I  gives  briefly  the  general  method  of  making  Re- 
connoissance;  Chapter  II  treats  of  Preliminary  Surveys;  while 
Chapter  III  relates  to  Location. 

Chapter  IV,  on  Transition-curves,  follows  the  method  adopted 
by  Professor  Crandall,  and  enables  one  to  locate  the  transition- 
curve  with  rigid  accuracy  where  such  is  necessary.  Approximate 
methods  are  also  given  by  means  of  which  the  curve  may  be  as 
easily  located  as  any  of  the  more  limited  easement  curves  ordi- 
narily met  with. 

Chapter  V,  on  Frogs  and  Switches,  contains  all  that  is  necessary 
for  their  location.  The  formulas  have  been  arranged  to  give  the 
desired  quantities  in  terms  of  the  frog  number  whenever  the  re- 
sulting equations  would  be  easier  of  application  than  the  trigono- 
metric ones  usually  given.  The  turnout  tables  are  unusually  full 
and  give  not  only  the  theoretical  lead  but  the  stub  lead  as  well, 
from  which  the  practical  lead  can  be  at  once  found  when  the 
length  of  switch-rail  is  known. 

Chapter  VI,  on  Construction,  tells  how  to  set  slope-stakes,  and 
gives  simple  methods  for  computing  areas  and  volumes  either 
directly  or  by  the  use  of  tables.  A  short  table  of  prismoidal 
corrections  is  given  for  end  sections  level,  and  also  a  formula  for 
three-level  sections,  by  means  of  which  a  suitable  table  may  be 
computed  if  desired. 

The  tables  at  the  end  of  this  book  have  been  arranged  with  a 
view  to  ease  of  reference,  for,  whatever  the  character  of  the  text, 
the  chief  value  of  a  field-book  must  depend  upon  the  ease  with 
which  the  tables  may  be  consulted  and  upon  their  extent  and 
accuracy.  Table  IX — Functions  of  a  One-degree  Curve — sepa- 
rates the  logarithmic  functions  on  the  one  side  from  the  natural 
functions  on  the  other  and  will  be  of  assistance  in  locating  these 
tables.  Table  XVI — Transition-curve  Table — reading  lengthwise 
of  the  page,  likewise  serves  to  separate  the  trigonometric  tables 
from  the  miscellaneous  tables  that  follow. 

Some  engineers  object  to  the  use  of  logarithmic  tables  in  the 
field,  but  for  them  the  natural  functions  are  at  hand ;  while  for 
those  who  prefer  logarithms  the  five-place  tables  of  logarithmic 
sines,  cosines,  etc.,  will  be  found  easy  to  consult  and  interpolate 
between. 


PREFACE,  V 

All  trigonometric  tables  are  five-place,  and  others  were  carried 
to  as  many  decimal  places  as  their  cliaracter  demanded. 

Tables  I,  III,  IV,  and  V  have  been  computed  to  agree  ^vith 
the  definition  of  the  degree  of  curve  requiring  curves  sharper 
than  7°  to  be  run  with  chords  less  than  100  feet  in  length,  as 
described  in  the  text.  Tables  XVII  and  XV 111  were  also  com- 
puted expressly  for  this  book. 

Tables  VI  and  XXVll  are  from  electrotyj^es  fram  Ca' iiart's 
Field  Book  for  Cioil  FJngineers  and  were  furnished  by  (linn  k  Co. 
Electrotypes  of  Tables  II.  X,  Xll,  XIII,  XIX,  XX,  XXIV,  XXV, 
XXVI,  and  also  XVI  —  this  last  being  from  Crundall's  book, 
'I'he  Transition  Curve — were  furnished  by  John  Wiley  &  Sons. 

Of  the  others,  some  were  arranged  from  standard  tables  and 
others  adapted  in  part  and  extended  to  increase  their  usefulness. 

It  will  be  noticed  that  vertical  lines  have  been  omitted  wher- 
ever practicable,  thus  rendering  it  easier  to  refer  to  the  tables. 

Acknowledgments    are    due    my    associate.   Professor  D.    W. 

Jpence,  for  aid  in  making  the  tabular  computations  and  in  reading 

proof. 

J.  C.  Nagle. 

College  Station,  Texas,  May,  1897. 


PREFACE  TO   THE  SECOND  EDITION. 


In  this  edition  some  of  the  typographical  and  other  minor  errors 
that  appeared  in  the  first  edition  have  been  eliminated.  Tables 
XXVIII  and  XXIX  have  been  added  in  order  to  increase  the  use- 
fulness of  the  book,  and  are  from  electrotypes  of  tables  in  Traut- 
v/ine's  Pocket  Book.  A  suggestion  has  been  made  by  one  who 
has  had  occasion  to  use  the  tables  quite  freely  that  Table  XIX  be 
extended  so  as  to  give  quantities  for  variations  of  one  tenth  of  a 
foot  in  center  heights,  but  such  extension  would  have  increased 
the  size  of  the  book  unduly.  When  closer  approximations  are 
wanted  than  are  given  by  Table  XIX  the  area  for  the  given  center 
height  can  be  taken  from  Table  XVII  and  by  entering  Table  XX 
irith  this  as  argument  the  quantity  can  be  at  once  read  off.  For 
«enter  heights  greater  than  those  given  in  Table  XVII  we  may 
refer  to  books  devoted  exclusively  to  earthwork  computations. 

J.  C.  N. 

College  Station,  Texas,  Jauuary,  1899. 


CONTENTS. 


CHAPTER  I. 

RECONNOISSANCE. 
Article  1.    Objects  of  Reconnoissance — How  Made. 

SECTION  PAGE 

1.  Relative  Importance  of  tlie  Work  of  Reconnoissance  and  Location..  1 

2.  Object  of  Reconnoissance 2 

3.  The  Instruments 2 

4.  Use  of  Maps , 4 

5.  Making  the  Reconnoissance 4 

CHAPTER  II. 

PRELIMINARY   SURVEYS. 

Article  2.    Objects;  The  Field  Corps;  Duties  of  the  Chief. 

6.  Objects  of  Preliminary  Surveys  6 

7.  The  Exploration-line .-  ^ 

8.  Data  Sought  in  Making  Preliminary  Surveys 7 

9.  The  Field  Corps , 7 

10.  The  Chief  of  Party,  Duties  of 7 

Article  3.    The  Transit  Party, 
a.    duties  of  the  members. 

11.  Composition  of  the  Transit  Party fi 

12.  The  Transitman 8 

13-17.  Other  Members  of  the  Party  £ 

18.  Instruments S 

B.     TRANSIT  ADJUSTMENTS— THE  VERNIER. 

19.  Kind  of  Transit S 

20.  To  Adjust  the  Plate  Levels IC 

21.  Parallax IC 

22.  To  Adjust  the  Line  of  Collimation IC 

23.  To  Adjust  the  Standards 11 

vii 


Vlll  CONTENTS. 

SECTION  PAGE 

24.  To  Adjust  the  Level  on  Telescope 12 

25.  Direct  and  Retiograde  Verniers  13 

26.  The  Least  Count  of  a  Vernier  c....o 13 

27.  To  Read  a  Vernier 14 

C.  ACCESSORIES. 

(1°)  The  Gradienter. 

28.  Description  and  Method  ofUsing  Gradienter 14 

(2°)  The  Stadia,  or  Telemeter. 

29.  Principle  of  the  Stadia 15 

30.  Formula  for  Line  of  Sight  Horizontal 15 

31 .  Formulas  for  Line  of  Sight  Inclined 16 

32.  The  Instrumental  Constant,  To  Find .^ IT 

33.  Reducing  the  Notes 17 

D.  FIELD-WORK. 

34.  station  Numbers , 18 

35.  Hubs  or  Plugs 18 

36.  Reference-points    . .   18 

37.  Alignment 18 

38.  Form  of  Transit  Notes , 19 

39.  Stadia  Methods  for  Preliminary  Surveys 19 

E.      OBSTACLES  IN  TANGENT. 

41.  To  Pass  an  Obstacle  by  Means  of  Parallel  Lines 20 

42.  To  Pass  an  Obstacle  by  Angular  Deflections 20 

43.  To  Measure  across  a  River 21 

Article  4.    The  Level  Party. 

44.  Make-up  and  Instruments 23 

45.  Work  of  the  Leveler  23 

46.  Work  of  the  Rodman 23 

adjustments  of  the  level. 

47.  To  Adjust  the  Line  of  Collimation  ...   23 

48.  To  Adjust  the  Level-bubble 24 

49.  To  AdjntJt  the  Wyes 25 

B.      THEORY  OF  LEVELING. 

50.  True  and  AuDarent  Level 25 

51.  The  Error  Due  to  Curvature 25 

52.  The  Difference  of  Elevation  of  Two  Points 26 

C.      FIELD-WORK. 

53.  The  Datum 27 

54  Bench-marks 27 

55  Work  in  the  Field 28 


\ 

CONTENTS.  ix 

SECTION  ,  PAGE 

56.  Tbe  Level  Notes 28 

57.  Precautions  when  Using  Level 29 

58.  TheRod /v;..^.. .,... 29 

Article  5,    The  Topographic  Party. 

59.  Instruments  Used ;  Area  to  be  Mapped 30 

80.  Methods  of  Recording  Data 30 

61 .  Topographers'  Field-sheets o . « 31 

62.  Use  of  the  Slope-level 31 

63.  Cross-section  Rods 32 

64.  The  Transit  and  Stadia  in  Topographical  Surveying 32 

Article  6.    Preliminary  Estimates. 

66.  Map  of  Preliminary  Lines . . 32 

67.  The  Profile 33 

68.  Preliminary  Estimates  of  Quantities 33 

69.  Report  of  the  Locating  Engineer 34 


CHAPTER  III. 

LOCATION. 

Article  7.    Projecting  Location. 

70.  Problems  Involved  in  the  Paper  Location 35 

71.  Hints  Regarding  Methods  of  Projecting  the  Line  35 

72.  Tbe  Curve-protractor 36 

r^'i.  Work  in  the  Field 37 

Article  8.    Simple  Curves. 

A.    definitions  and  formulas. 

74.  Definitions 38 

75.  To  Find  the  Radius  R,  the  Degree  of  Curve  Being  Known 40 

76.  To  Find  the  Length  of  Curve 42 

77.  The  Functions  of  a  One-degree  Cui've 42 

79.  To  Find  Z), -R  and  C  Being  Known 43 

80.  To  Find  the  Tangent  Distance  T,  I  and  B  Being  Known 43 

81.  To  Find  R,  Given  I  and  T 44 

82.  Given  i  and  D,  to  Find  the  Long  Chord  i.C 44 

83.  Ordinates  from  Chord 45 

84-86.  To  Find  the  External  E 48 

87.  To  Find  7?,  E  and  2  Given 49 

88.  To  Find  r,  S  and  i  Given 49 

89.  To  Find  the  Deflection  Offset  from  Chord  Produced 49 

90.  To  Find  the  Tangent  Deflection  Offset 50 

91 .  The  Sub-tangential  Deflection  Offset 51 

92.  To  Find  the  Tangent  Offset  z 52 

■33.  Difference  in  Length  of  Arc  and  Long  Chord  53 


X  CONTENTS. 

B       LOCATING  SIMPLE  CURVES. 
SECTION  PAGE 

94.  To  Locate  a  Curve  vvitli  the  Chain  by  Offsets  from  Chords  ProduceU  55 

95.  To  Locate  a  Curve  by  Offsets  from  Tangent 57 

96.  To  Locate  a  Curve  by  Offsets  from  a  Long  Chord 58 

97.  To  Locate  a  Curve  with  Transit  and  Chain 59 

98.  The  Index-angle 60 

99.  Subdeflection-angles 60 

100- lOL  Transit  Notes 61 

C.      OBSTACLES. 

102.  To  Pass  an  Obstacle  on  a  Curve 63 

103.  To  Locate  a  Curve  when  the  P.  C.  is  Inaccessible , 64 

104.  To  Pass  to  Tangent  when  the  P.  T.  is  Inaccessible  67 

105-107.  To  Pass  a  Curve  through  a  Given  Point 69 

108.  To  Locate  a  Tangent  to  a  Curve  from  an  Outside  Point .     71 

109.  To  Run  a  Tangent  to  Two  Curves  of  Contrary  Flexure 78 

D.      CHANGE  OF  LOCATION. 

110.  To  Locate  a  Curve  Parallel  to  a  Given  Curve 73 

111.  To  Change  P.O.  in  Order  to  Make  P.T.  Fall  in  a  Parallel  Tangent. .      74 
112   To  Change  R  and  P.C   to  make  P.T.  Fall  in  Parallel  Tangent,  on 

Same  Radial  Line 75 

113.  To  Find  Change  in  P.C.  or  R  for  a  Given  Change  in  7 76 

114.  Required  the  Change  in  P.C.  and  R  for  a  Given  Change  in  7,  the 

P.T'  Unchanged     .    ...      . ..     77 

115.  To  Find  Nevv  Radius  for  a  Given  Change  in  T 77 

116.  To  Find  New  72  to  Connect  P.C.  with  a  Parallel  Tangent 78 

Article  9.    Compound  Curves. 

A.    location  problems. 

117.  Given  Both  Tangents  and  One  Radius,  to  Find  the  Other  Radius ...    80 

118.  Given  One  Radius,  the  Long  Chord  and  the  Angles  it  Makes  with 

Tangents,  to  Find  the  Other  Radius  and  Central  Angles     .     82 

119.  Given  the  Radii  and  Central  Angles,  to  Find  the  Tangents,  the  Long 

Chord, and  the  Angles  it  Makes  with  Tangents ,.... 82 

120.  Given  the  Long  Chord  and  Angles  Made  with  Tangents,  to  Find 

Both  Radii  when  Common  Tangent  is  Parallel  to  Long  Chord S3 

B.    obstacles. 

121.  To  Locate  Second  Branch  when  P.  C.  is  Inaccessible 84 

C.     CHANGE  OF  LOCATION. 

122.  To  Compound  a  Simple  Curve  so  P.T.  shall  Fall  in  a  Parallel  Tan- 

gent..  .     85 

123.  To    Find  Change  in  P.CC.  Necessary  to  xMake  P.T.  Fall  in  a  Par- 

allel Tangent 86 

124.  To  Change  P.CC.  and  Second  Radius  so  P.T.  shall  Fall  in  a  Par- 

allel Tangent,  on  Same  Radial  Line  , 89 


CONTENTS.  XI 

SECTION  PAGE 

125.  To  Chanjje  P.C.C.  and  Second  Radius  to  Cause  P. T,  to  Fall  at  a 

New  Point  in  Same  Tangent  91 

126.  To  Substitute  a  Three-centered  Compound  Curve  for  a  Simple  One.    94 

127.  To  Substitute  a  Curve  for  a  Tangent  Uniting  Two  Curves 95 

Article  10.    Track  Problems. 

128.  Reversed  Curves,  Where  to  Use 96 

129.  To  Connect  a  Located  Curve  with  an  Intersecting  Tangent 97 

130.  To  Locate  a  Y 100 

131.  A  Reversed  Curve  between  Parallel  Tangents 102 

132.  A  Crossover  between  Parallel  Tracks  when  a  Fixed  Length  of  Tan- 

gent is  Inserted , 105 

133.  A  Reversed  Curve  with  Unequal  Angles 106 

134.  A  Reversed  Curve  between  Fixed  Points  106 

135.  To  Connect  Two  Divergent  Tangents  by  a  Reversed  Curve 107 

136.  To  Change  P.jR. a  so  P.T.  shall  Fall  in  a  Parallel  Tangent.. 108 

137.  To  Find  the  Radius  of  a  Curved  Track 109 


CHAPTER  IV. 

TRANSITION-CURVES. 

Article  11.    Theory  of  the  Transition-ccrve. 

138.  Elevation  of  Outer  Rail  on  Curves 110 

139    Requirements  of  the  True  Transition-curve Ill 

140.  Notation  Employed ill 

141.  Equation  of  Transition-curve 112 

142.  Transition-curve  Angle,  / 114 

143.  Coordinates  of  Points 114 

144.  Deflection-angles 115 

145.  Explanation  of  Transition-curve  Tables 118 

146.  To  Unite  the  Brandies  of  a  Compound  Curve  by  a  Transition- 

curve  119 

147.  Length  of  Transition-curve  to  be  Taken 121 

Article  12.    Field-work. 

A.    field  formulas. 

148.  When  to  Use  the  Simplified  Formulas 122 

149.  Simplified  Formulas  for  Transition-curves 122 

1.50.  Offsets 124 

151.  Compound^^  Curves 125 

B.    setting  out  transition-curves. 

153.  Location  by  Offsets 125 

154.  Location  i;y  Deflection-angles  126 

155.  Form  of  Transit  Notes  for  Tran.sition-curves 128 


.  i 


Xll  CONTENTS. 

Article  13.    Transition  curve  Problems. 

SECTION  PAGE 

156.  Tangent  Distances  and  Exterual  for  Equal  Offsets 129 

157.  Tangent  Distances,  Offsets  Unequal 130 

158.  Transition-curves  Inserted  without  Changing  the  Vertex  of  Cir- 

cular Curve 131 

159.  Transition-curves  Inserted  with  Least  Deviation  from  Old  Track....  133 

160.  Transition-curves  Inserted   at  Ends  of  Long  Circular  Curve,  Cen- 

tral Portion  Undisturbed  133 

161.  Transition-curve  Inserted  at  P.C.C.  by  Changing  Radius  of  Second 

Branch 136 

162.  To  Insert  Transition-curves  at  the  Ends  of  Two  Circular  Curves 

United  by  a  Common  Tangent 138 

163.  To  Unite  a  Tangent  and  Circular  Curve  when  the  Offset  Cannot  be 

Directly  Measured   . 139 

164.  Inserting  Transition-curves  in  Old  Track 140 

165.  Remarks  on  Tabular  Interpolations  140 


CHAPTER  V. 

FROGS  AND  SWITCHES. 

Article  14.  Turnouts. 

A.  turnouts  from  straight  lines.^ 

166.  Definitions 143 

167.  To  Find  the  Lead,  I,  and  Radius,  R,  in  Terms  of  the  Frog  Number, 

N,  and  Gauge,  g 144 

168.  Given  R  and  g,  to  Find  N,  I,  and  Frog-angle,  F 146 

169.  To  Find  Theoretic  Length  of  Switch-rail 146 

170.  To  Find  Lead  and  Number  of  Crotch-frog  for  a  Double  Turnout  to 

Opposite  Sides  of  Main  Track 147 

171.  To  Find  Turnout  Radius  and   Lead  of  Crotch-frog  in  Terms  of 

Crotch-frog  Number 148 

172.  To  Find  Radius  of  Curve  from  Point  of  Middle  Frog  to  Point  of 

Main  Frog,  Given  i\r,,  N,  andN' 148 

173.  Double  Turnout  to  Same  Side  of  Main  Track 150 

174.  To  Find  Radius  of  Curve  between  Frog-points  for  a  Double  Turn- 

out to  Same  Side  of  Main  Track 151 

175.  To  Unite  Main  Track  with  Siding.  Reversing  Point  Opposite  Frog  . .  152 

176.  To  Lay  Out  a  Ladder-track  .. 153 

B.    turnouts  from  curves. 

177.  To  Find  Lead  and  Radius  for  Turnout  to  Concave  Side  of  Main 

Line 154 

178.  To  Find  Lead  and  Radius,  Turnout  to  Convex  Side 157 

179.  To  Find  Theoretic  Length  of  Switch-rail <, 158 

180.  To  Unite  Main  Track  with  a  Concentric  Siding 160 


CONTENTS.  Xlll 


C.     THE  STUB  LEAD, 
SECTION  PAGE 

181.  Definitions 162 

182.  Given  N,  t,  and  g,  to  Find  the  Stub  Lead 362 

183.  Turnout  Table  and  Explanation 163 

184.  To  Stake  Out  a  Turnout 165 

185.  Curving  Rails 165 

Article  15.    Crossovers. 

186.  Crossover  between  Parallel  Straight  Tracks,  a  Tangent  between 

Frog-points 166 

187.  A  Crossover  in  the  Form  of  a  Reversed  Curve 168 

188.  A  Crossover  with  Fixed  Length  of  Intermediate  Tangent 168 

189.  A  Crossover  between  Curved  Main  Tracks 168 

Article  16.    Crossing-frogs  and  Crossing-slips. 

A.  crossing-frogs. 

191.  Length  of  Rail  Intercepted  between    Two  Intersecting   Straight 

Tracks 170 

192.  Angles  of  a  Set  of  Crossing  frogs,  One  Track  Curved 170 

193.  Angles  of  a  Set  of  Crossing-frogs,  Both  Tracks  Curved 171 

B.  crossing-slips. 

195.  Length  and  Radii  of  Slip-rails,  Both  Tracks  Straight 172 

196.  Length  and  Radii  of  Slip-rails,  One  Track  Curved  172 

197.  Length  and  Radii  of  Slip-rails,  Both  Tracks  Curved! 173 


CHAPTER  VI. 

CONSTRUCTION, 

Article  17.    Definitions  ;  General  Considerations  ;  Vertical 
Curves  ;  Elevation  op  Outer  Rail. 

199.  The  Division  Engineer 176 

200.  The  Resident  Engineer 176 

201-204.  Definitions 177 

205.  To  Find  the  Grade-point,  Longitudinal  Slope  Uniform 178 

206.  Vertical  Curves , 17? 

207.  Elevation  of  Outer  Rail  on  Curves 188 

208.  Easing  Grade  on  Curves 183 

Article  18.    Earthwork. 

A.      SETTING  slope-stakes. 

209.  The  Distance  Out  for  Level  Sections ISS 

210.  To  Find  Position  of  Slope-stakes  for  Surface  Inclined 184 

211.  Cross-section  Notes •. i86 

212.  Irregular  Sections 187 

Jil3.  Staking  Out  Openings 187 


XIV  CONTEJSTS. 

SECTION  PAGE 

214.  Manner  of  Marking^  Stakes , 187 

215.  Shrinkage— Growth 187 

216.  Borrow-pits,  Drainage  of,  etc 188 

B.      AREAS  OF  SECTIONS. 

218.  Area  of  Three-level  Section.  188 

219.  Area  of  Five-level  Section 189 

220.  General  Formula  for  Areas 190 

221.  Explanation  of  Table  of  Areas  of  Level  Sections  and  the  Three- 

level  Correction , 191 

C.      VOLUME  OF  EARTHWORK. 

222.  Where  Cross-sections  should  be  Taken 192 

223.  Volume  by  Averaging  End  Areas 192 

224.  The  prismoidal  Formula 193 

225.  Form  of  Record 195 

226.  The  Prismoidal  Correction 195 

227.  Computation  of  Volumes  when  Passing  from  Cut  to  Fill 198 

228.  Use  of  Tables  of  Volumes  in  Making  Preliminary  Estimates 199 

229.  Side  Ditches 199 

230.  Earthwork  on  Curves 199 

231.  Overhaul .-.201 

Article  19.    Grade  and  Ballast  Stakes,  Culverts,  Bridge.^, 

and  tunnils. 

232.  Grade  and  Center  Stakes 202 

233.  Ballast-stakes 202 

235.  Openings  of,  for  Culverts,  Trestles,  etc 202 

236.  Bridge  Piers  and  Abutments 203 

237.  Tunnels , 204 

Article  20.    Monthly  and  Final  Estimates. 

238.  Monthly  Estimates 205 

239.  Measurements  for  Earthwork 206 

240.  Classification  of  Earthwork 206 

211.  The  Progress  Profile 207 

242.  Masonry  Estimates 207 

243.  Bridge  Estimates 207 

244.  Track  Material ."* 207 

245.  Blank  Estimate  Sheets 208 

246.  Monthly  Payments 208 

247.  Extras » 208 

248.  Final  Estimate 208 

249.  Acceptance 209 

TABLES. 

Table  Showing  Length  of  Transition -curve  to  be  Taken  121 

Table  of  Values  of  g  -  Vgi  for  Stub  Lead 163 

Turnout  Table 164 

Table  of  Corrections  for  Vertical  Curves , 181 


CONTENTS.  XV 

PAGE 

Tabie  of  Elevation  of  Outer  Rail  on  Curves  182 

Table  of  rrisinoi'dal  Corrections  for  Level  Sections 196 

I.  Radii  of  Curves 212 

II.  Minutes  in  Decimals  of  a  Degree 215 

III.  Tangential  Offsets  216 

IV.  Long  Chords  and  Actual  Arcs 217 

V.  Mid-ordinates  to  Long  Chords 218 

VI.  Logarithms  of  Numbers , .  220 

VII.  Logarithmic  Sines  and  Cosines 238 

VIII,  Logarithmic  Tangents  and  Cotangents 253 

IX.  Functions  of  a  One-degree  Curve 268 

X.  Natural  Sines  and  Cosines  298 

XI.  Natural  Secants  and  Cosecants 307 

XII.  Natural  Tangents  and  Cotangents 320 

XIII.  Natural  Versines  and  Exsecants 332 

XIV.  Coordinates  for  Transition-curves 355 

XV    Deflection-angles  for  Transition-curves 356 

XVI.  Transition-curve  Table  358 

XVII.  Areas  of  Level  Sections 371 

XVIII.  Corrections  for  Three-level  Ground 375 

XIX.  Cubic  Yards  per  100  ft.  in  Terms  of  Center  Height 376 

XX.  Cubic  Yards  per  100  ft.  in  Terms  of  Sectional  Area 382 

XXI.  Rise  per  Mile  of  Various  Grades 386 

XXII.  Slopes  for  Topography 387 

XXIII.  Material  Required  for  One  Mile  of  Track 387 

KXIV.  Mutual  Conversion  of  Feet  and  Inches  into  Meters  and  Centi- 
meters      388 

XXV.  Mutual  Conversion  of  Miles  and  Kilometers 389 

XXVI.  Length  of  V  Arc  of  Latitude  and  Longitude 389 

XXVIL  Trigonometric  and  Miscellaneous  Formulas 390 

XXVIII.  Square  Roots  and  Cube  Roots  of  Numbers  from  .1  to  28  395 

XXIX.  Squares.  Cubes,  Square  Roots,  and  Cube  Roots,  of  Numbers 

from  1  to  1000 396 


A    FIELD-MANUAL    FOR    RAILROAD 

ENGINEERS. 


CHAPTER  I. 

RECONNOISSANCE. 

Article  1.     Objects  op  Reconnoissance — How  Made. 

1.  The  question  of  the  selection  of  the  proper  route  for  a  Hue 
of  railway  is  essentially  an  economic  one,  luvolving  not  only  the 
cost  of  construction,  but  of  maintenance  and  operation,  and  a 
consideration  of  the  immediate  and  future  traffic  likely  to  pass 
over  the  completed  road. 

The  engineer  upon  whom  devolves  the  duty  of  making  the 
surveys  for  a  railroad  is  not  often  called  upon  to  determine 
whether  it  should  or  should  not  be  built,  though  his  preliminary 
estimate  may  decide  those  whose  duty  it  is  to  do  so  :  the  problem 
confronting  him  is  how  to  secure  the  best  line,  answering  a  given 
purpose,  for  the  least  cost.  Keeping  in  mind  the  proper  working 
of  the  completed  road,  the  problem  may  be  divided  into  two  gen- 
eral parts  : 

First.  The  selection  of  the  general  route  between  terminal 
points,  and  in  some  cases  the  selection  of  the  terminals  them- 
selves. 

Second.  The  fitting  of  the  line  to  the  ground  in  such  a  manner 
as  will  render  the  cost  of  constructing  and  operating  the  road  a 
minimum. 

The  first  is  by  far  the  more  important  and  difficult  operation, 
requiring  the  highest  grade  of  engineering  skill — a  fact  too  sel- 
dom recognized  by  those  selecting  engineers  for  this  work.  The 
acquirement  of  the  necessary  skill  can  result  only  from  1oT)<r 
practice  and  close  observation,  coupled  with  the  ability  to  !ul;y 


2  A    FIELD-MANUAL   FOR   RAILROAD    EXGINKEIIS. 

grasp  and  weigh  all  the  complex  features  of  the  question.  A 
passiug  reference  only  can  be  made  to  it  in  this  little  volume, 
which  is  intended  to  furnish  hints  and  aids  to  the  better  execution 
of  the  second  part.  For  the  benefit  of  the  beginner  who  has  to 
do  with  the  location  and  construction  a  few  definitions  and  hints 
relating  to  reconnoissauce  will  be  given  before  going  on  to  the 
special  problems  arising  in  the  work  of  the  railroad  engineer. 

2.  The  Reconnoissance  is  a  rapid,  general  survey  of  the  area 
through  which  the  proposed  railroad  must  pass,  made  only  with 
such  instruments  as  can  be  easily  carried,  and  which  should  ena- 
ble the  engineer  to  restrict  the  more  accurate  instrumental  work 
that  follows  to  one  or  two  general  lines.  The  time  required  for 
this  part  of  the  work  will  in  geneial  be  only  a  small  fraction  of 
the  time  consumed  in  location,  involving  the  service  of  very  few 
men;  yet  there  is  no  part  of  the  work  more  rapidly  and  im- 
properly done — not  always  because  tbe  engineer  in  charge  under- 
estimates its  importance,  but  because  he  is  not  usually  allowed 
sufficient  time  in  which  to  study  thoroughly  the  area  under  con 
sideralion. 

Properly  the  reconnoissance  includes  the  determination  of  the 
terminal  points  of  the  road,  but  the  locating  engineer  is  usually 
relieved  from  the  necessity  of  selecting  these  points,  and  the 
question  reduces  to  that  of  finding  the  best  available  line  which 
admits  of  being  built,  maintained,  and  operated,  at  the  least  cost 
beticeen  two  given  points. 

The  reconnoissance  must  be  made  over  an  area — not  a  line  or 
lines.  Even  what  seems  the  most  unpromising  portion  should 
be  carefully  studied,  for  the  engineer  can  never  be  satisfied  he 
has  selected  the  best  route  until  he  has  convinced  himself  by  care- 
fnl  study  that  all  others  are  inferior.  Too  much  haste  on  reron- 
noissance  means  either  a  poor  line  or  a  much  greater  expenditure 
of  time  and  money  on  the  preliminary.  No  amount  of  notes  or 
topography  can  take  the  place  of  an  intimate  personal  knowledge 
of  the  problems  to  be  encountered,  and  hence  the  reconnoissance 
and  preliminary  survey  should  be  made  by  the  engineer  who  is 
to  locate  the  road. 

3.  The  Instruments  needed  will  rarely  be  more  than  a  pocket- 
compass,  hand-level,  aneroid  barometer,  field-glasses,  and  some- 
times a  pedometer  or  an  odometer. 

{a)  The  Pocket-compass  is  used  to  obtain  the  magnetic  bear- 
ini;s  of  lines  and  the  angles  they  make  with  each  other. 


RECONNOISSANCE.  3 

(b)  The  Hand-level  enables  one  to  obtain  differences  of  ele- 
vation between  points  not  far  apart. 

(c)  The  Aneroid  Barometer  gives  approximate  lieights  of  the 
mercury  column,  and  serves  to  roughly  determine  the  difference 
of  elevation  of  given  points.  In  addition  to  the  scale  giving 
readings  in  inches,  it  should  have  also  a  scale  graduated  to  give 
readings  in  feet.  If  two  aneroids,  which  have  been  previously 
compared,  are  read  simultaneously,  one  at  each  of  the  points 
whose  difference  of  elevation  is  desired,  or  if  the  same  aneroid  is 
read  at  each  successively  at  a  short  interval  of  time,  during  which 
the  atmospheric  pressure  has  not  sensibly  altered,  we  may  find 
the  difference  of  elevation  by  thes  formula* 

d  =  60000  (log  H-  log  70(l  +  ^\l~  ^^),     .     .     (1) 

in  which  d  is  the  difference  of  altitude  in  feet,  H  and  h  the 
barometric  readings  in  inches — the  logarithms  being  of  the  com- 
mon or  Briggs  kind,  J' and  t  the  temperatures  of  the  two  stations 
in  Fahrenheit  degrees. 

If  the  sum  of  the  temperatures,  T-{- 1,  is  taken  as  105°,  formula 
(1)  reduces  to 

(Z  =  63000  (logs'- log  7i) (1') 

Example. — The  reading  of  the  barometer  at  the  foot  of  a 
mountain  is  28.8  inches,  and  at  the  top  26.7  inches.  Required 
the  height  of  the  mountain. 

By  (1').  d  =  63000  (log  28.7  -  log  26.7)  =  2071  feet. 

The  effect  of  temperature  on  the  metal  of  the  instrument 
should  be  considered  in  the  barometric  formula  when  very  pre- 
cise work  is  to  be  done  ;  but  this  correction,  being  small,  may  be 
neglected  in  the  rough  work  of  reconnoissance,  particularly  since 
the  makers  of  the  instrument  construct  it  in  such  a  way  as  to 
compensate,  as  closely  as  possible,  for  such  changes  of  tem- 
perature. 

{d)  The  Pedometer  is  an  instrument  which  automatically 
counts  the  number  of  steps  made  by  a  person  when  the  instru- 
ment is  attached  to  his  belt  ;  then,  knowing  the  average  length 
of  step,  the  distance  passed  over  can  be  readily  computed. 

The  Odometer  registers  the  nu'nber  of  revolutions  of  a  wheel 
ta  which  it  is  attached,  and  tlie  number  of  revolutions  multiplied 
by  the  circumference  of  tlie  wheel  gives  the  space  passed  over.       « 

*  See  Plympton's  Aneroid  Barometer,  p.  88,  for  formula  (1). 


4  A    FlELI>-:iIANUAL   FOR   RAILROAD    ENGINEERS. 

4.  The  Map. — Before  beginning  the  reconnoissance  the  engi- 
neer should  provide  himself  with  the  best  available  map  of  the 
region  to  be  traversed  ;  if  this  is  a  topographic  one,  he  can  at 
once  determine  from  it  the  lines  that  are  likely  to  justify  an 
examination  ;  and  even  if  it  is  only  a  sketch-map,  he  can  get 
material  assistance  by  observing  the  courses  of  the  streams  and 
remembering  that  their  positions  indicate  the  relative  elevations 
of  the  portion  of  the  region  through  which  they  flow.  Thus  the 
large  streams  follow  the  lines  of  least  elevation,  and  the  manner 
in  which  the  lateral  streams  unite  with  the  principal  one  indi- 
cates the  general  trend  of  the  terrain.  Two  streams  flowing 
nearly  parallel  approacli  or  recede  from  each  other  according  as 
the  intervening  land  diminishes  or  increases  in  altitude.  Two 
streams  flowing  away  from  each  other  on  opposite  sides  of  a 
divide,  and  having  their  source  therein,  approach  each  other 
closest  at  the  point  of  least  elevation,  and  indicate  the  position  of 
a  pass  or  the  lowest  point  of  the  dividing  ridge.  The  study  of 
any  good  contour  map  covering  sufficient  area  will  illustrate  the 
laws  governing  the  courses  followed  by  streams. 

The  elevations  of  a  few  correctly  mapped  points,  when  obtain- 
able, from  the  map  or  otherwise,  serve  as  a  guide  in  tentatively 
fixing  on  the  maximum  gradient  to  be  employed  and  the  amount 
of  development  needed. 

A  skillful  engineer  will  thus  be  enabled  to  project  his  lines 
with  sufficient  accuracy  to  enable  him  to  select  on  the  ground  the 
most  feasible  route  or  routes  for  his  preliminaries  in  the  least 
possible  time.  He  should  guard  against  the  conviction,  however, 
that  it  is  unnecessary  for  him  to  look  elsewhere  than  along  the 
projected  routes  ;  for  the  inaccuracies  of  the  map,  local  peculiari- 
ties, the  nature  of  the  excavation  and  embankment,  the  number 
and  cost  of  bridges  and  other  mechanical  structures, — all  these 
may  conspire  to  make  the  most  promising  map-line  inferior  to 
some  other  whose  advantages  have  to  be  sought  for  on  the 
ground. 

6.  Having  tentatively  decided  on  the  limiting  grades  and  cur- 
vature to  be  employed,  the  engineer  goes  carefully  over  the 
ground,  examining  the  entire  area  that  seems  likely  to  afford 
passage,  in  order  to  determine  whether  a  suitable  line  may  l)e 
secured  for  the  grades  and  curves  previously  assumed.  With  his 
pocket-compass  he  takes  the  bearings  of  lines,  and  by  means  of 
the  hand-level  and  aneroid  determines  differences  of  elevation. 


RECONNOISSANCE.  O 

Distances  are  estimated  by  the  eye,  paced,  and  tlie  count  taken 
from  the  pedometer,  or,  if  the  country  admits  of  the  use  of  a 
vehicle,  talceu  from  the  odometer  readings.  If  a  well-gaited 
saddle-horse  is  used,  very  good  results  may  be  gotten  by  timing 
him,  or  by  the  use  of  the  pedometer  if  his  stride  is  uniform. 

But  in  all  cases  much  dependence  must  be  placed  on  the  ability 
to  estimate  with  the  eye  differences  of  elevation  and  distances. 
The  ability  to  do  this  with  even  reasonable  accuracy  comes  only 
from  long  practice  and  careful  observation,  even  to  the  most 
gifted  in  this  respect.  New  and  unexpected  conditions  some- 
times deceive  even  the  most  practiced  eye,  but  under  ordinary 
conditions  almost  any  one  can  train  his  eye  to  estimate  horizontal 
distances  fairly  well.  Vertical  heights  are  more  deceptive,  pos- 
sibly because  we  have  less  practice  in  this  line,  and  the  mind 
seems  naturally  to  exaggerate  the  vertical  as  compared  with  the 
horizontal  ;  practice,  however,  will  enable  us  to  make  allowance 
for  the  natural  tendency  to  overestimate  heights  and  slopes. 

The  ground  should  be  gone  over  in  both  directions,  for  the  ap- 
pearance may  be  quite  different  when  approached  from  different 
quarters.  Ruling  points,  such  as  a  pass  in  the  mountains,  the 
crossing  of  a  large  stream,  or  a  town  or  city  through  which  the 
road  must  be  built,  serve  to  reduce  the  problem  to  a  number  of 
special  ones,  each  having  its  own  solution. 

In  a  mountainous  region  offering  a  limited  number  of  possible 
routes,  but  heavy  construction  work,  it  may  often  happen  that 
the  location  of  a  line  is  a  much  less  difficult  operation  than  in  an 
open,  rolling  country  offering  a  score  of  possible  lines,  between 
which  the  engineer  making  the  reconnoissance  must  decide, 
selecting  only  those  that  in  his  judgment  seem  to  justify  an 
accurate  instrumental  survey. 

The  engineer  must  keep  constantly  in  mind  all  the  factors  of 
the  general  problem  of  economic  location  and  maintenance,  and 
successful  operation  of  trains.  One  line  may  cost  more  for  con- 
struction and  maintenance  than  another,  but  less  for  operation, 
or  may  invite  less  traffic.  In  all  cases,  however,  the  question 
of  grades,  curvature,  length  of  line,  earthwork,  and  mechanical 
structures  are  the  controlling  elements  to  be  considered. 

Having  decided  upon  the  route  or  routes  over  which  to  run 
preliminaries,  these  are  marked  on  the  map,  and  the  engineering 
party  organized  and  put  in  the  field,  with  all  the  necessary 
instruments. 


CHAPTER  II. 
PRELIMINARY  SURVEYS. 

Article  2.    Objects;  The  Field  Corps  ;  Duties  of  the  Chief. 

6.  The  Objects  of  the  preliminary  surveys  are  to  secure  all  the 
data  necessary  to  determine  wliicli  one  of  the  routes  selected  on 
reconuoissance  is  the  most  feasible,  all  things  considered,  and  the 
approximate  cost  of  construction.  In  rough  country  it  will  be 
economical  to  make  two,  or  even  three,  surveys  over  the  route  se- 
lected for  location  before  beginning  to  place  the  line  in  the  yjosition 
it  is  finally  to  occupy.  The  first  of  these  is  often  omitted,  and  is 
called  an  "exploration-line  "  ;  it  will  frequently  save  the  making 
of  the  more  expensive  "preliminary"  over  one  or  more  of  the 
routes. 

7.  The  Exploration-line  may  be  made  with  either  transit  or 
compass,  and  consists  of  a  rapidly  run  line,  made  for  the  purpose 
of  determining  the  maximum  curvature  and  gradients  with  which 
to  project  the  preliminary.  It  will  not  be  necessary  to  make  a 
detailed  study  of  the  region  at  this  time,  the  distances  and  eleva- 
tions, with  such  sketch  topography  as  may  be  easily  taken,  being 
all  that  is  needed.  The  magnetic  bearing  of  lines  is  taken  by 
the  compassman,  and  the  chainmen  align  each  other  with  the  flag 
set  by  the  flagman.  As  the  progress  of  the  level  party  will  be 
slower  than  that  of  the  compass  party,  it  will  be  economical  to  add 
an  extra  rodman,  and  sometimes  a  recorder.  The  compassman 
may  sketch  in  the  features  adjacent  to  the  line  while  waiting  for 
his  chainmen,  who  may  be  either  in  front  of  or  behind  the  com- 
pass. 

The  stadia  method  of  surveying — to  be  spoken  of  later — would 
seem  to  offer  exceptional  advantages  for  this  work — only  three  or 
four  men  being  needed  in  addition  to  the  chief.  With  it,  by  set- 
ting the  transit  over  alternate  stations,  very  rapid  progress  may  be 
made,  and  obstacles  avoided  with  as  much  or  greater  ease  than 
with  the  compass. 

The  exploration-line  will  more  than  pay  for  itself  in  showing 

6 


PRELIMINARY   SURVEYS.  Y 

what  routes  it  will  be  unnecessary  to  make  preliminaries  over, 
and  in  indicating  the  most  feasible  one.  It  should  be  run  over  all 
the  routes  selected  on  reconnoissance. 

8.  The  Preliminary  Survey  follows  the  exploration,  or,  when 
this  is  omitted,  comes  next  after  the  reconnoissance.  It  may,  with 
advantage,  be  made  in  two  parts — first  and  second  preliminary. 
It  is  made  with  such  instrumental  accuracy  as  the  nature  of  the 
case  may  demand,  sufficient  data  being  obtained  to  determine  the 
best  line  on  which  to  locate  and  the  approximate  cost  of  construc- 
tion. •  The  rapidity  with  which  this  work  can  be  done  will  depend 
on  the  care  with  which  the  reconnoissance  was  made.  The  pre- 
liminary line  should  approximate,  as  closely  as  the  eye  can  deter- 
mine, to  the  position  the  located  line  should  occupy,  and  forms  the 
base  on  which  the  topographic  work  rests.  In  reasonably  easy 
country,  where  exploration-lines  have  been  run,  one  preliminary 
should  suffice  for  each  route,  but  in  difficult  regions  it  will  be  best 
to  run  a  second  preliminary.  If  portions  of  the  route  are  easy,  fol- 
lowed by  difficult  parts,  it  will  often  be  sufficient  to  "back  up" 
and  re-run  the  difficult  portion  until  a  reasonably  satisfactory  line 
has  been  obtained. 

9.  The  Field  Corps  consists  of  a  chief  of  party,  transitman, 
leveler,  rodman,  two  chainmen,  rear  rodman  or  "back-flag," 
stakeman,  and  two  or  more  axemen.  If  a  topographic  party  is 
added,  as  it  should  Ije  in  any  but  the  easiest  country,  there  will  be 
also  a  topographer  with  two  or  more  assistants.  A  cook  and 
teamster  will  be  needed  with  the  camp  outfit. 

The  corps  is  usually  divided  into  the  following  parties  : 

(a)  The  Transit  Party. 

(h)   The  Level  Party. 

(c)  The  Topographic  Party. 

10.  The  Chief  of  Party  receives  his  orders  from  the  chief  en- 
gineer, or  such  other  officer  as  may  be  in  charge,  directs  the  mo- 
tions of  the  surveying  corps,  and  is  responsible  for  their  conduct 
and  progress.  He  provides  accommodations  and  supplies,  pays  all 
expenses,  taking  receipts  or  vouchers  for  all  outlays — in  dupli- 
cate when  required.  In  the  less  thickly  populated  sections  he 
must  provide  tents,  wagons,  cook,  and  all  necessary  camping  outfit 
and  supplies.  He  must  direct  the  field  oj^erations  in  person,  keep- 
ing in  advance  of  the  transit,  establish  turning-points  or  hubs, 
and  direct  the  transitman  in  the  proper  course.     He  should  keep 


8  A   FIELD-MAlS^rAL   FOR   RAILROAD    ENGINEERS. 

a  record— or  direct  the  transitman  and  topographer  to  do  so — of 
the  character  of  earthwork  likelv  to  be  encountered,  the  places 
where  drains,  culverts,  bridges,  cattle-guards,  etc.,  are  needed; 
the  nature  of  material  for  embankment,  piling,  etc.,  adjacent  to 
the  line  ;  the  probable  amount  of  clearing  and  grubbing,  and  all 
other  features  likelv  to  affect  the  cost  of  construction.  He  should 
see  that  the  names  of  property  owners  and  residents  along  the 
line  and  the  positions  and  bearings  of  property  lines,  when 
possible,  are  noted. 

He  should  have  authority  to  discharge  assistants — except  transit- 
man,  leveler,  and  topographer — whose  services  are  unsatisfactory, 
and  in  many  cases  it  will  be  best  for  him  to  have  entire  control, 
engaging  or  discharging  any  member  of  the  corps  as  circumstances 
may  require. 

Article  3.     The  Transit  Party. 
A.  Duties  of  the  Members. 

11.  The  Transit  Party  should  consist  of  a  transitman,  head 
chainman,  rear  chainman,  rear  flagman,  stakeman,  and  as  many 
axemen  as  may  be  required — rarely  less  than  two  even  for  open 
country. 

12.  The  Transitman  cares  for  his  instrument,  keeping  it  in  ad- 
justment; directs  the  chainmen  into  line;  notes  the  angle  between 
successive  tangents  as  read  on  plates;  notes  also  the  bearings  of 
tangents,  of  highways,  streams,  and  property  lines  (on  location), 
with  the  plus  at  which  the  line  crosses  them.  If  there  is  no 
topographic  party  he  must  make  sketches,  on  the  right-hand  page 
of  note-book,  of  the  surface  features  adjacent  to  the  line;  the 
red  Hue  down  the  middle  of  page  represents  the  transit  line, 
whether  straight,  broken,  or  curved,  to  which  the  sketches  are 
adjusted.  He  must  see  that  the  axemen  keep  in  line,  in  order 
that  no  unnecessary  chopping  may  be  done.  Large  trees  need 
rarely  be  felled  on  preliminary,  even  when  a  given  general  course 
has  to  be  followed,  for  small  angles  may  be  turned  to  avoid  them, 
the  deflections  to  right  being  made  to  approximately  balance  those 
to  left. 

When  the  chief  of  party  is  absent  the  transitman  is  ranking 
man.  and  will  take  temporary  charge. 

13.  The  Head  Chainman  carries  a  range-pole  or  "flag,"  and 
drags  the  chain,    which  he  must  see  is  straight  and  horizontal 


PRELIMIXARY    SURVEYS.  9 

when  setting  a  point  for  a  stake.  He  directs  the  stakeman  where 
to  drive  his  stake,  calling  out  the  number  after  the  rear  chainman 
has  read  and  called  out  the  number  on  his  stake;  he  keeps  the 
axemen  in  line  by  setting  his  flag  and  going  ahead,  directing  them 
where  to  cut  bv  keeping  them  in  line  with  the  flag  and  transit. 
The  speed  of  the  party  is  dependent  on  the  rapidity  and  accuracy 
with  which  he  can  set  his  flag  in  position,  by  ranging  with  stakes 
already  set  between  him  and  transit,  and  in  seeing  that  the 
axemen  make  all  their  work  count. 

14.  The  Rear  Chainman  must  be  careful  to  hold  his  end  of 
the  chain  in  the  proper  place,  and  that  it  is  kept  straight  and  taut 
when  the  head  chainman  is  setting  a  stake.  He  must  give  all 
pluses,  note  the  number  on  each  stake  as  he  comes  up  to  it,  and 
see  that  the  stakeman  has  marked  it  correctly;  he  must  make  a 
note  of  pluses  for  roads,  fences,  streams,  etc.,  to  be  given  to  the 
transitman  later  on. 

15.  The  Stakeman  must  keep  himself  supplied  with  stakes 
about  1^"  X  2'  X  ~^",  marking  the  number  on  them  plainly,  and 
drivino^  them  as  directed  bv  the  head  chainman. 

If  sawed  stakes  are  not  provided,  he  must  cut  the  stakes  and 
face  them  for  the  numbers.  He  must  keep  on  hand  a  number  of 
plugs  or  "hubs,"  to  be  driven  flush  with  the  ground  and  having 
the  point  where  flag  rested  marked  with  a  tack.  About  ten  or 
twelve  inches  to  the  left  of  and  facing  the  hub  a  guard  stake  is 
driven,  on  which  is  marked  the  station  number,  and  which  enables 
one  to  find  the  hub  at  any  time. 

16.  The  Axemen  do  all  necessary  clearing  and  chopping  in 
order  that  the  transit  and  level  parties  may  have  a  clear  sightway, 
and  yet  restrict  the  work  of  clearing  to  a  minimum.  One  of  them 
may  be  detailed  to  keep  the  stakeman  supplied  with  stakes. 

17.  The  Rear  Flagman  holds  his  flag  on  the  last  turning- 
point  for  the  transitman  to  use  in  back-sighting. 

18.  The  Instruments  used  by  the  party  are  the  transit  (or 
compass),  one-hundred-foot  chain  or  tape,  range-poles,  and  the 
necessary  axes  and  hatchet  for  axemen  and  stakeman. 

B.    Transit  Adjustments — The  Vernier. 

19.  For  railroad  work  the  transit  is  usually  plain,  but  it  is 
often  convenient  to  have  a  clamp  and  tangent  movement  to  tele- 


10        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

scope,  a  vertical  circle,  a  level  on  telescope,  stadia  wires,  and  a 
gradienter;  the  solar  attachment  will  rarely  be  needed. 

20.  To  Adjust  the  Plate  Levels.  — The  axis  of  the  instrument 
is  set  at  right  angles  to  the  plates  by  the  manufacturer,  so  that 
when  the  axis  is  made  vertical  the  plates  will  be  horizontal. 

In  making  adjustments  remember  that  a  complete  reversal 
always  doubles  any  existing  error. 

Place  the  bubble-tube  parallel  to  a  diagonal  pair  of  leveling- 
screws,  and  bring  the  bubble  to  the  centre  of  its  run.  Revolve 
the  instrument  180°  on  the  vertical  axis,  and  the  level- tube  will 
be  parallel  to  the  same  pair  of  leveling-screws  as  before,  but 
reversed.  If  the  bubble  has  moved  from  its  central  position 
bring  it  /t«(/'-way  back  by  means  of  the  capstan-headed  screws  at 
the  ends  of  the  tube.  Relevel  and  repeat  until  the  bubble  remains 
at  the  centre  after  reversal.  Do  the  same  for  the-  other  bubble. 
Both  bubbles  should  remain  at  the  centres  of  their  tubes  during  a 
complete  reversal. 

21.  Parallax  is  an  apparent  movement  of  the  cross- wires  with 
respect  to  the  object  sighted  when  the  eye  is  moved  from  side  to 
side  of  the  eyepiece,  and  shows  that  the  image  does  not  fall  in  the 
plane  of  the  cross-wires.  In  precise  measurements  it  should  be 
removed  before  making  an  observation  with  the  telescope.  To  do 
this,  first  bring  the  cross-wires  clearl}'  into  view,  when  the  object, 
glass  is  turned  towards  the  sky,  then,  when  sighting  an  object, 
note  if  there  is  any  relative  movement  of  cross-wires  and  image 
when  the  eye  is  moved  from  side  to  side  at  the  eyepiece  ;  if  there 
is,  refocus  the  object-glass  until  this  movement  disappears. 

22.  To  Adjust  the  Line  of  OoUimaiion  is  to  make  the  line 
joining  the  intersection  of  cross-wires  and  optical  center  of  objec- 
tive describe  a  plane  perpendicular  to  the  horizontal  axis  of  instru- 
ment. 

First  Method. — Level  the  instrument  and  clamp  the  move- 
ments on  vertical  axis.  Sight  some  well-defined  object  distant 
about  the  length  of  an  average  sight,  and  in  the  same  horizontal 
plane  as  telescope.  Reverse  the  telescope  on  its  horizontal  axis, 
and  fix  a  point  about  as  far  from  instrument  as  first  point,  and  in 
the  same  horizontal  plane.  Revolve  the  instrument  on  its  vertical 
axis  and  sight  the  first  point;  then  reverse  the  telescope  and  note 
if  line  of  sight  cuts  the  second  point.  If  not,  loosen  the  capstan- 
headed  screws  holding  cross- wire  ring  and  move  the  vertical  wire 


PRELIMIXARY    SURVEYS.  11 

over  one  fourth  tLe  apparent  error— since  tLere  were  two  reversals 
— remembering  that  the  image  of  the  cross- wires  is  inverted,  while 
that  of  the  object  appears  in  its  true  position.     Test  by  repetition. 

Second  Method. — If  the  limb  graduations  can  be  relied  on 
they  may  be  used  in  adjusting  the  vertical  wire.  With  the  instru- 
ment level  sight  a  well-defined  point,  then  revolve  180°  by  vernier- 
plate,  reading  both  verniers;  reverse  telescope,  and  note  if  line  of 
sight  cuts  the  point.  If  not,  correct  one  half  the  apparent  error  by 
moving  diaphragm  ;  then  test  by  repetition. 

The  manufacturers  adjust  the  object-glass  slide  so  that  the  ob- 
jective travels  in  the  telescope  axis,  and  this  adjustment  is  not 
liable  to  serious  derangement.  It  is  well,  however,  to  sometimes 
test  by  adjusting  the  line  of  collimation  for  both  near  and  distant 
objects.  If  not  correct  for  both,  move  the  ring  which  guides  the 
rear  end  of  object-glass  slide  until  the  adjustment  is  correct  for 
both  positions. 

Next  make  the  vertical  wire  vertical  by  noting  if  it  coincides 
throughout  its  length  with  a  plumb-line,  or  by  observing  if  it  de- 
viates from  a  point,  on  which  the  intersection  has  been  fixed,  when 
the  telescope  is  elevated  or  depressed.  Any  error  is  corrected  by 
turning  the  ring  after  slightly  loosening  the  screws  holding' it. 

The  horizontal  wire  should  also  be  adjusted  so  that  the  inter- 
section of  the  cross-wires  will  be  in  the  axis  of  the  telescope  ;  if 
the  transit  is  to  be  used  as  a  leveling  instrument  this  adjustment 
is  essential. 

Drive  a  stake  close  to  the  instrument,  and  with  the  telescope 
clamped  as  nearly  horizontal  as  can  be  conveniently  done  read  a 
rod  held  on  top  of  the  stake  ;  about  800  feet  distant,  and  in  line 
with  first  stake  and  instrument,  drive  a  second  stake  and  read  the 
rod  on  it.  Revolve  180°  on  vertical  axis,  reverse  the  telescope  and 
bring  the  horizontal  wire  to  the  former  reading  when  the  rod  is 
held  on  first  stake  ;  if  the  reading  on  the  second  stake  is  not  the 
same  as  before,  correct  one  Jialf  the  apparent  error  by  moving  the 
cross-wire  ring.  Repeat  as  a  test.  The  vertical  wire  should  again 
be  tested  lest  the  movement  of  the  ring  may  have  caused  it  to 
change. 

23.  To  Adjust  the  Standards  is  to  make  the  plane  described 
by  the  line  of  collimation  vertical.  Set  up  the  transit  about  as  far 
in  front  of  some  high  building,  or  other  tall  object,  as  the  highest 
])oint  that  can  be  sighted  is  above  the  base.  Level  the  instrument 
and  fix  the  intersection  of  the  cross-wires  on  the  highest  point  that 


12        A   FIELD-MANUAL   FOR  RAILROAD   ENGINEERS. 

can  be  easily  sighted.  Depress  tlie  telescope  and  fix  a  point  near 
tlie  base  of  the  building  at  about  the  beiglit  of  the  telescope.  Un- 
clamp  and  revolve  on  the  vertical  axis  until  the  telescope  reversed 
cuts  the  lower  point.  Clamp  the  plates  and  raise  the  telescope 
until  the  cross-wires  are  at  the  height  of  the  upper  point.  If  they 
cut  it  the  standards  are  in  adjustment.  If  they  do  not,  bring 
them  half-way  back  by  means  of  the  adj  ustable  screws  at  the  top 
of  one  of  the  standards.     Repeat  as  a  test. 

24.  To  Adjust  the  Level  on  Telescope  is  to  make  the  bubble 
stand  at  the  center  of  its  run  when  the  line  of  sight  is  horizontal. 
Bring  the  telescope  as  nearly  horizontal  as  may  be  convenient,  and 
take  readings  on  the  tops  of  two  pegs  in  the  same  vertical  plane 
with,  and  equidistant  from,  the  instrument — say  300  feet.  The 
difference  of  readings  will  equal  the  difference  of  elevation  of  the 
pegs;  this  difference  may  be  obtained  with  the  wye-level  if  pre- 
ferred. 

Move  the  instrument  to  a  point  beyond  one  of  the  pegs  and  in 
line  with  both.  Set  up  as  close  to  nearer  peg  as  convenient,  but 
not  so  close  that  the  rod  cannot  be  easily  read.  Bring  the  tele- 
scope as  nearly  horizontal  as  possible,  and  read  on  both  pegs.  If 
the  difference  of  readings  equals  their  difference  of  elevation  the 
line  of  sight  is  horizontal,  and  the  bubble  may  be  brought  to  the 
center  by  means  of  the  adjustable  screws  attaching  the  level-tube 
to  the  telescope.  If  this  is  not  the  case,  we  must  set  the  telescope 
so  the  reading  on  second  peg  equals  the  reading  on  first  peg  plus 
the  difference  of  elevation  ;  then  read  again  on  first  peg  and  pro- 
ceed  as  before  until  the  condition  is  satisfied.  Or  we  may  proceed 
as  follows : 

In  Fig.  1  let  the  transit  be  at  0,  and  A  and  B  be  the  pegs.  AC 
is  a  horizontal  through  A,  so  that  CB  is  the  difference  of  elevation 


Fio.  1. 


of  A  and  B.     Suppose  line  of  sight  to  cut  the  rods  at  ^and  7), 
we  must  find  DO  so  that  the  target  may  be  set  at  the  proper  read- 


rilELlMlNARY    SURVEYS.  13 

ing  to  make  tlie  line  of  sight  horizontal.     Let  0F=  a,  FG  =  b, 
EA  =  r,  DB  =  r',  CB  =  k.     Draw  DH  parallel  to  CA  and  OG; 
then  ER—T-\-k-r'. 
From  similar  triangles 

DG  =  ER-^  ={r^k-r')[~J-j 

Set  the  target  at  a  reading  GB  =  GB  H-r',  sight  to  G,  and  tl.:^ 
line  of  sight  will  be  horizontal.    Bring  the  bubble  to  the  center  ol 
its  run  while  the  telescope   is  in   this   position,  and  the  adjust 
ment  is  complete. 

If  desired,  a  correction  for  the  curvature  of  the  earth  and  re- 
fraction may  be  introduced,  but  for  short  sights  this  is  a  useless 
refinement. 

25.  The  Vernier  is  an  auxiliary  scale  for  measuring  smaller 
divisions  than  those  graduated  on  the  limb.  There  are  two 
classes,  the  direct-reading  and  the  retrograde,  according  as  the 
fractional  parts  of  limb  readings  are  taken  on  that  side  of  the 
zero  of  vernier  scale  towards  which  the  vernier  has  moved  with 
respect  to  the  limb,  or  the  reverse.  On  the  direct  vernier  a  cer- 
tain number  of  divisions  on  the  vernier  equals  the  same  number 
of  divisions  on  the  limb,  less  one  ;  on  the  retrograde  there  is  one 
more  division  on  limb  than  on  vernier  when  the  same  space  is 
covered  by  both. 

26.  The  Least  Count  of  a  vernier  is  the  smallest  subdivision  of 
limb  graduation  that  can  be  read  by  it,  and  equals  the  difference 
of  one  space  on  limb  and  one  on  vernier. 

Let  I  =  value  of  one  space  on  limb  ; 

V  =  value  of  one  space  on  vernier  ; 

n  =  number  of  spaces  on  vernier. 
Then  for  the  direct  vernier 

nv  =  {n  —  V)l ; 

from  which  we  get  the  least  count, 

n 
For  the  retrograde  vernier 

nv  =  {n  +  1)1, 


14        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

from  which  the  least  count  is 

V  —  I  =  —, 
n 

the  same  result  as  found  for  the  direct  vernier. 

So,  to  find  the  least  count  :  Divide  the  value  of  one  limb  space  by 
the  number  of  spaces  on  the  vernier. 

For  example  :  If  the  limb  of  a  transit  is  divided  to  half-degrees 
and  the  number  of  spaces  on  the  vernier  is  30,  the  least  count 
will  be  \  divided  by  30,  or  -^^  of  a  degree — that  is,  1  minute. 

27.  To  Read  a  Vernier,  take  the  number  of  the  last  division  on 
limb  back  of  the  vernier  zero,  then  look  along  the  vernier  until  a 
line  is  found  to  coincide  with  a  line  on  the  limb  ;  add  the  number 
of  this  vernier  line,  multiplied  by  the  least  count,  to  the  scale 
reading,  and  the  result  will  be  the  required  reading. 

C.  Accessories. 
(V)  The  Gradienter. 

28.  The  Gradienter  consists  of  a  tangent-screw  having  a 
micrometer-head,  attached  to  one  of  the  standards  of  the  transit 
and  capable  of  being  clamped  to  the  horizontal  axis  of  the  tele- 
scope. It  is  used — as  its  name  indicates — in  running  grades,  and 
it  accurately  measures  a  small  vertical  angle  in  terms  of  its  tan- 
gent. The  screw  is  so  cut  that  one  revolution  moves  the  tele- 
scope through  an  angle  whose  tangent  at  one  hundred  feet  from 
the  instrument  has  a  certain  value,  usually  one  foot.  The  grad- 
uated head  is  divided  into  100  parts,  so  that  one  division  corre- 
sponds to  0.01  ft.  at  100  feet  from  instrument. 

To  run  a  given  gradient,  bring  the  telescope  level  and  read  the 
micrometer-head  of  screw;  then  turn  the  screw  as  many  divisions 
as  there  are  hundredths  of  a  foot  rise  or  fall  in  100  feet,  and  with 
"  target  set  at  the  height  of  the  horizontal  axis,  points  on  the 
surface  corresponding  to  the  given  grade  can  be  found. 

For  example  :  To  run  a  0.75  per  cent  grade,  move  the  microm- 
eter milled  head  75  graduations  from  the  horizontal. 

When  used  as  a  Telemeter,  we  may  either  measure  the  space 
on  the  rod  moved  over  by  the  line  of  sight  for  a  given  number  of 
revolutVms  of  the  screw,  or  we  may  note  the  number  of  revolu- 
ticms  required  to  move  the  line  of  sight  over  a  certain  space  on 
rod.  The  second  method  is  the  more  accurate,  particularly  for 
long  sights. 


PRELIMINARY    SURVEYS. 


J5 


(3°)  The  Stadia,  or  Telemeter. 

29.  The  Stadia  is  an  instrument  for  determining  the  distance 
of  a  point  from  the  observer  by  noting  the  space  intercepted  on  a 
rod  by  a  given  visual  angle,  as  determined  by  two  auxiliary  wires 
parallel  to,  and  equidistant  from,  the  horizontal  wire  of  the  transit 
telescope.  When  used  with  an  ordinary  leveling-rod  the  wires 
should  be  adjustable;  if  they  are  fixed  (which  for  some  reasons 
is  preferable),  the  rod  must  be  graduated  to  correspond.  In 
addition  to  the  distance  of  a  point  from  the  instrument,  the  differ- 
ence of  elevation  is  determined  by  observing  the  angle  made  by 
line  of  sight  with  the  horizontal  when  the  middle  horizontal  wire 
cuts  a  point  on  the  rod  as  high  above  the  ground  as  is  the  centre 
of  the  telescope. 

The  horizontal  position  of  the  point  is  determined  from  its 
magnetic  bearing,  or  the  azimuth  of  line  of  sight  with  reference 
to  some  fixed  line,  usually  the  north-south  line. 

30.  Line  of  Sight  Horizontal. — In  Fig.  2  let  a  and  h  be  the 
stadia  wires,  AB  the  intercept  on  the  rod.     The  secondary  axes 

A 


Fig.  2. 


aA  and   hB   pass   through   the   optical   center  0.     Let   h  =  ah, 
r  =  AB,  d  —  distance  of  cross-wires  from  objective,  Z>  —  distance 
of  rod  from  objective. 
From  similar  triangles. 


From  optics. 


In  which/ is  the  focal  length  of  objective. 
Eliminating  d  from  these  two  equations. 


h 

r 

d 

~D 

d^  D 

i 
"  f 

16        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

Ijet  c  be  tlie  mean  distance  of  objective  from  center  of  instru- 
mentf.  Adding  this  to  I)  gives,  for  the  distance  of  the  rod  from 
the  center  of  the  instrument. 


l  =  c+f+£r. 


f 

J-  may  be  made  constant,  when  (2)  becomes 

ft 


I  =  a  -\-  kr. 


(2) 


(2') 


31.  Line  of  Sight  Inclined. — When  the  line  of  sight  is  not 
level  it  is  difficult  to  hold  the  rod  perpendicular  thereto  ;  hence 
the  rod  is  held  vertical,  the  angle  of  inclination  measured,  and  a 
correction  applied.     In  Fig.  3 


-Vr) 


B 


let  r  ■■ 
r'  : 

H: 
V 

n  ■ 


Fig.  3. 

CD  be  the  reading  on  rod  held  vertical  ; 

FE,  the  reading  perpendicular  to  line  of  sight  ; 

AG,  the  horizontal  distance  from  ^  to  ^  ; 

BG,  the  difference  of  elevation  between  A  and  B ; 

BAG,  the  angle  of  inclination  of  line  of  sight. 


Assume  angles  AFB  and  AEB  =  90^  from  which  they  rarely 
differ  more  than  15'  to  17'.     Then,  since  FBC  =  n^ 


r  z=  r  cos  n. 


PRELIMINARY    SURVEYS.  1 


1^ 


By  (2').  AB  =  a-{-  kr'. 

Hence  AB  =  a  -{-  kr  cos  n. 

From  triangle  ABG 

H  =  AB  cos  n 

.'.  n  =  a  cos  n-\-kr  cos'  n (3) 

F=  AB  sin  n\ 
.  •.  F  =  a  sin  n  -\-  kr  sin  n  cos  n. 

But  2  sin  71  cos  w  =  sin  2n. 

Hence  V  ■=  a  sin  n-\-  ^kr  sin  2n. (4) 

32.  The  Instrumental  Constant  a  [—  c  +/  of  (2)]  may  be 
found  by  measuring  the  distance  from  center  of  instrument  to 
mean  position  of  objective,  which  equals  c  ;  then  focusing  on  a 
very  distant  object,  preferably  a  star,  and  measuring  from  center 
of  objective  to  plane  of  cross-wires,  which  equals/.  The  sum  of 
these  distances  is  a  in  formulas  (3)  and  (4). 

If  the  stadia  wires  are  fixed,  k  may  be  found  by  measuring  for- 
ward on  level  ground  the  distance  a  from  plumb-line,  and  from 
this  point  a  further  distance  h ;  then  note  carefully  the  stadia 
reading  r  when  the  telescope  is  level.     Then,  remembering  (2)', 

a-\-'b  =  a  ■\-  kr. 

.'.  k  =  —,  a.  constant  ratio. 
r 

If  the  stadia  wires  are  adjustable,  we  may  so  adjust  k  that  any 
desired  reading  may  be  had  for  a  given  length  of  base.  A  con- 
venient value  of  k  is  100,  which  corresponds  to  an  intercept  of 
1  foot  on  the  rod  at  100  feet  from  a  point  a  feet  in  front  of  the 
instrument,  2  feet  at  200  feet  in  front,  etc. 

33.  A  Stadia  Table  based  on  formulas  (3)  and  (4)  4s  published 
by  the  D.  Van  Nostrand  Company  in  Winslow's  Stadia  Surveying, 
and  can  be  used  more  rapidly  than  the  formulas.  Johnson's  Re- 
duction Diagram,  by  John  Wiley  &  Sons,  gives  values  of  i/and  V 
graphically.  Colby's  S/ide-ride,  manufactured  by  Mahn  &  Co., 
St  Louis,  gives  values  of  y  for  distances  in  feet,  yards,  or  meters 
to  tenths  of  a  foot,  and  can  be  used  with  great  rapidity. 


18        A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


D.  Field-work. 

34.  Station  Numbers  should  begin  with  zero  for  the  initial 
stake,  and  are  marked  on  rear  side  of  stake,  from  the  top  down- 
ward, the  number  of  the  preliminary,  A,  B,  G,  etc.,  being  marked 
on  the  forward  side.  The  marking  should  be  with  kiel,  or  crayon 
that  will  withstand  the  action  of  sun  and  rain.  Stakes  may  be 
set  every  hundred  feet  or  only  at  even  stations,  as  preferred. 

35.  Hubs,  or  Plugs,  are  transit  turning-points,  and  are  short, 
flat-topped  stakes  driven  into  the  ground  flush  with  the  surface. 
The  flag  is  held  on  the  top  and  carefully  aligned,  the  position  of 
the  point  being  marked  by  a  tack.  A  special  tack  with  concave 
bead  offers  a  foothold  for  point  of  flag  when  used  in  backsight 
ing.*  About  10  inches  to  the  left  of  and  with  numbered  sid' 
facing  the  hub  is  driven  a  guard-stake  to  mark  its  position. 

36.  Reference- points  are  two  or  more  hubs,  with  guard -stakes 
in  each    of   two  lines  making  a  good   intersection  angle  at  the 
point  whose  position  they  serve  to  locate.     They  should  be  driver, 
beyond   reach   of  disturbance,    and  are  used  in  replacing  a  dig 
located  hub. 

These  need  rarely  be  used  on  preliminary. 

37.  Alignment. — It  is  not  intended  that  the  preliminary  an(\ 
location  lines  occupy  exactly  tlie  same  position  ;  hence  consider- 
able latitude  is  allowable  in  the  size  and  number  of  angles 
turned,  care  being  taken,  however,  that  the  maximum  curvature 
need  not  be  exceeded  on  location.  Large  trees  and  other  obstruc- 
tions may  be  avoided  by  turning  a  SMiall  angle  until  the  obstacle 
has  been  passed,  then  making  a  deflection  in  the  opposite  sense. 
Bearings  of  tangents  are  taken  with  the  needle,  to  serve  as  a 
check  on  the  angle  read  on  the  plates. 

In  easy  country  not  requiring  a  topographic  party  large  angles 
should  not  he  turned,  a  succession  of  small  ones  with  short  inter- 
vening tangents  being  substituted  in  order  to  make  the  prelimi- 
nary profile  approximate  more  closely  to  the  location  profile. 
These  short  tangents  may  conveniently  be  the  long  chords  of  the 
curve  that  is  to  follow. 

*  ^  uch  a  tack  is  manufactured  by  the  A.  S.  Aloe  Co.,  St.  Louis. 


PRELININARY    SURVEYS. 


19 


38.  The  Transit  Notes  may  be  kept  in  the  form  below,  which 
shows  both  pages  of  the  note-book.  The  notes  run  from  the 
bottom  up,  the  right-hand  page  being  reserved  for  sketches  ;  the 
red  line  up  the  middle  of  the  page  represents  the  transit  line, 
whether  straight  or  broken,  to  which  the  sketches  must  be 
adjusted. 


Sta. 

Angle. 

Calculated 
Course. 

Magnetic 
Course. 

Remarks  and  Sketches. 

68 

1 

67© 

20°  0'  L. 

N.  1°  48'  W. 

N.  1°  45'  W. 

0 

66 

65 

64 

63© 

6°  2'  R. 

N.  IS"  12'  E. 

N.  18°  15'  E. 

0 

6-2 

61 

39.  Stadia  Methods  for  Preliminary  Surveys. — Preliminary 
lines  are  usually  run  with  the  transit,  but  the  compass  will 
answer  nearly  as  well  in  most  cases,  besides  admitting  of  more 
rapid  work.  The  transit  and  stadia  method  might  well  be  em- 
ployed, and  would  effect  considerable  saving  in  the  cost  of  pre- 
liminary surveys.  For  some  reason  railroad  engineers  have  not 
regarded  it  with  favor,  though  it  is  extensively  employed  in 
topographic  surveying  where  the  map  is  to  be  used  for  work  that 
is  often  more  precise  than  needed  for  railroad  preliminaries. 

Particularly  is  this  method  applicable  to  exploration  lines. 
With  the  transit  and  stadia  the  entire  surveying  corps  need  not 
exceed  five  or  six  men,  the  instrument  man  acting  as  transitman, 
leveler.  and  topographer  all  in  one.  The  only  objection  would 
seem  to  be  in  the  amount  of  reduction  the  notes  would  need ; 
however,  with  tables  and  slide-rule  (see  33)  this  work  may  be 
very  rapidly  done.  For  vertical  angles  of  less  than  one  degree 
the  horizontal  reduction  can  be  neglected,  and  with  side  readings 
for  topography  the  angle  may  be  5  or  10  degrees  without  necessi- 
tating the  correction.  Vertical  heights  are  found  by  the  slide- 
rule  or  by  charts. 

This  method  would  really  necessitate  the  making  of  a  topo- 
graphic map  along  a  narrow  strip  of  country,  from  which  the 
profile  could  readily  be  taken.  With  a  skilled  observer  and  two 
to  four  rodmen  the  progress  may  be  more  rapid,  and  fully  as 
good  for  the  purpose  intended  as  the  more  expensive  method 
usually  employed. 


30        A   FIELD-MANUAL    FOR    RAILROAD    EN'GIXEERS. 

The  transit  need  onlv  be  set  at  alternate  stations  (which  may  be 
any  length  within  the  reading  limits  of  the  wires),  the  bearings  to 
other  stations  and  points  off  the  line  being  taken  with  the  needle. 
The  horizontal  angle  should  also  be  read  on  the  plates  for  points 
on  stadia  line,  as  a  check  on  the  bearings. 

E.   Obstacles  in  Tangent 

40.  Obstructions  to  vision  and  measurement  in  tangent  may  be 
avoided  in  a  number  of  ways,  a  few  of  which  are  given  in  the 
following  problems.  Other  methods  of  avoiding  them  will  sug- 
gest themselves  in  special  cases. 

The  same  devices  may  be  used  on  location,  but  it  is  more  im- 
portant to  maintain  a  clear  sight  way  then  ;  so,  when  possible,  we 
should  remove  the  obstruction. 

41.  To  Pass  an  Obstacle  by  Means  of  Parallel  Lines. — In 
Fig.  4,  0  is  the  obstruction,  AB  the  obstructed  line.     At  B  set 


H 


A  B      ^^^^  C 

Fig.  4. 

transit  ;  turn  90^  and  measure  BF  long  enough  to  clear  obstruc- 
tion. Set  transit  at  F,  make  BFG  —  90\  and  measure  FG. 
Move  to  G  and  backsight  to  F,  making  FGC  =  90".  Measure 
GC  —  FB,  and  move  to  C,  where  the  angle  GCD  is  made  equal 
to  90°.     CD  is  the  desired  line,  and  BC  =  FG. 

Otherwise,  at  J.  and  B  erect  perpendiculars;  take  BF=AE; 
produce  EF,  and  at  G  and  H,  beyond  0,  erect  perpendiculars  mak- 
ing GG=  HD  —  FB.    CD  will  be  the  desired  line,  and  BG  =  FG. 

42.  To  Pass  an  Obstacle  by  Angular  Deflections. 
General  Case.  Angle  anything  less  than  90\ 
At  B  (Fig.  5)  on  the  obstructed  line  deflect  an  angle  a  to  one 
side  and  measure  BC,  taking  C  so  that  after  deflecting  2a  to  the 
other  side  CZ)  will  clear  the  obstruction.  Make  CD  =  BC  a-nd 
deflect  an  angle  a  to  the  same  side  as  at  B\  DE  will  lie  in  AB 
produced.     Draw  Cfi^  perpendicular  to  52);  then 

BD  =  BH4-  HD  =  2BC    cos  a (5) 


PEELIMIXARY   SURVEYS. 


21 


F.' 


Fig.  5. 

Example.— Suppose  a  =  14°  10',  BC  =  CD  =  520  ft. 

BD  =  2X  520  X  0.96959  =  1008.37  feet. 

Special  Case.     Afig/e  60  degrees. 

In  this  case  the  triangle  BBFiFig.  6)  is  equilateral  and  BF  = 

BD  =  DF. 

Should  it  be  inconvenient  to  run  to  D  we  may  stop  at  C,  having 
measured  BC.     At  C  deflect  60"  and  measure  CE;  at  E  again  de- 


Fig.  6. 
fleet  60'  and  make  EF=  BC.     At  i^  a  final  deflection  of  60'  in  the 
opposite  sense  will  put  the  telescope  in  the  desired  line,  FG^  and 

BF=BC-\-CE.     ......     (5a) 

43.  To  Pass  an  Obstmction,  such  as  a  River,  when  the  Pre- 
ceding Methods  are  Inapplicable. 
First  Case.     Point  beyond  obstruction  lymble. 
In  Fig.  7  let  BC  be  required. 


Fig.  7. 
At  B  erect  and  measure  the  perpendicular  BD  ;  set  instrument 
at  B  and  measure  angle  BBC  =  a  ;  then 

BC  =BDt&nii (6) 


52        A   FIELD-MAIfUAL   FOR   RAILROAD   ENGINEERS. 

Or,  if  a  trigonometric  table  is  not  at  band,  make  CDE ^=  90"  and 
fix  the  point  E  where  DE  intersects  AB  ;  measuring  EB  there 
results,  from  similar  triangles, 

CB_BD 

BD  ~  EB' 


BD'^ 
whence  CB  =-^T7r (6«) 

Otherwise,  if  a  right  angle  at  B  is  not  convenient,   measure 
angles  CBD  =  b,  BDC  =  a,  and  side  BB.    Then  c  =  180°-  (a+t). 
From  triangle  BBC, 

BC  =  BD^^.       ......     (6&) 

sm  c 

Example.— «  =  66\h=  70°,  BB  =  400  feet. 

By  {6h),  BC  =  400  ^!"  ^^o  =  409.8  feet. 

•^        '  sm  54 

Second  Case.     Point  beyond  obstrniction  invisible. 
At  B  (Fig,  8)  measure  angle  b  and   line  BE ;  move  to  E  and 
measure  angle  y,  and  set  hubs  on  line  EG  so  the  line  BC  will  pass 


Fig.  8. 
between  them.     Angle  z  =  ECB  =  180  -  (&  +  y).    Then  from  tri- 
angle BEC 

BG=BE^^ (7) 

sm  2 

Produce  EB  to  B,  where  BC  will  be  sure  to  clear  obstruction; 
measure  BB. 

From  triangle  BBC, 

tan  K^  -  ^)  ^  BC-  BB 

t&n  {{a -\-j')       BC-\-BB 

But  a  -{-x  =:b,  hence 

^       ^.  ,       BC-BB    ^       .,  ,^. 

tan  lia  -  a-)  =  ^^  -  .  tan  ^6.         ...     (8) 


PRELIMINARY    SURVEYS.  23 

The  sum  and  difference  of  a  and  x  are  now  known,  so  botli  may 
be  readily  found. 

At  D  set  off  the  angle  a  with  the  transit,  and  have  the  chainmen 
stretch  a  cord  between  the  hubs  set  on  line  EC  Sit  C.  Now  signal 
the  flagman  to  move  his  rod  along  this  cord  until  the  vertical  wire 
cuts  it  at  C.  Set  a  hub  here  and  place  the  transit  over  it.  Sight 
to  D  or  E,  reverse  telescope  and  deflect  into  CII. 

Article  4. — The  Level  Party. 

44.  The  Level  Party  consists  generally  of  two  members,  the 
Jeveler  and  a  rodman ;  sometimes  an  axeman  is  added  to  keep  the 
lodman  supplied  with  pegs  for  turning-points  and  in  clearing  the 
hue  of  sight  for  the  level.  As  the  party  follows  the  transit  little 
or  no  clearing  will  be  needed.  The  instruments  used  are  a  level, 
a  rod,  and  a  hand-axe  or  hatchet. 

45.  The  Leveler  makes  all  necessary  observations  with  his 
instrument,  keeping  a  neat,  accurate  record  of  readings  and  ele- 
vations ;  also  the  positions  and  elevations  of  benches  and  turning- 
points.  He  should  work  out  elevations  of  stations  while  the  rod- 
man  is  going  from  one  station  to  the  next ;  he  must  see  that  the 
rodman  gives  him  readings  at  points  where  the  longitudinal  slope 
changes  suddenly,  recording  the  plus.  He  must  plot  his  profile 
at  night,  or  at  such  times  as  the  chief  of  party  is  likely  to  need  it. 
The  rodman 's  readings  at  turning-points  should  be  checked. 

46.  The  Rodman  holds  his  rod  at  each  station,  calling  out  the 
number.  If  stakes  are  set  only  at  even  stations,  he  must  hold  his 
rod  midway  between  stakes,  the  point  being  found  by  pacing  the 
distance.  Target-readings  need  only  be  taken  at  turning-points 
and  benches,  and  the  rodman  should  keep  a  record  of  these  in 
his  "peg-book,"  checking  the  calculations  of  leveler  for  heights 
of  instrument  and  elevations  of  turning-points.  At  any  marked 
surface  change  he  will  hold  his  rod,  calling  out  the  plus  to  leveler. 
He  must  assist  the  leveler  in  plotting  up  the  notes. 

A.  Adjustments  of  the  Level. 

47.  To  Adjust  the  Line  of  Collimation  is  to  bring  the  inter- 
section of  the  cross-wires  into  the  optical  axis  of  the  telescope. 

Set  up  and  level  the  instrument,  then  bring  the  vertical  wire 
into  coincidence  with  a  plumb  line  or  vertical  edge  of  a  building, 


24        A    FIELD-MAXUAL    FOR    RAILROAD    ENGINEERS. 

at  the  mean  leugth  of  sight,  and  note  if  the  vertical  wire  is  truly 
parallel  thereto.  If  it  is  not,  loosen  the  capstan -headed  screws 
holding  cross-wire  ring  and  turn  slightly  so  that  the  wire  is 
parallel  to  the  vertical  line. 

Loosen  the  wye-clips  and  bring  the  vertical  wire  into  coin- 
cidence with  the  line  and  clamp  the  instrument.  Rotate  the 
telescope  in  the  wyes  180°  and  note  if  the  wire  coincides  with  the 
line.  If  not,  correct  07ie  half  the  error  by  loosening  one  and 
tightening  the  opposite,  of  the  capstan-headed  screws  that  hold 
the  cross-wire  ring  in  place,  remembering  that  the  image  of 
the  cross  wires  is  inverted  by  the  eyepiece. 

Turn  the  telescope  until  the  horizontal  wire  is  parallel  to  the 
plumb-line  or  edge  of  building,  and  make  the  same  test  and 
correction.  Repeat  for  both  wires.  The  horizontal  wire  is  the 
one  on  which  the  accuracy  of  leveling  depends,  but  it  is  wise  to 
have  both  adjusted.  Their  intersection  should  remain  on  a  point 
during  a  complete  rotation  of  the  telescope  in  the  wyes. 

48.  To  Adjust  the  Level-bubble  is  to  bring  the  axis  of  IhQ 
level-tube  into  the  same  vertical  plane  with  the  line  of  collimation, 
and  to  make  the  bubble  stand  at  the  center  when  the  line  of  sight 
is  horizontal. 

Since  the  axis  of  the  telescope  coincides  with  the  line  joining 
the  center  of  the  wye-rings  (which  requires  these  to  be  of  the 
same  size),  it  is  sufficient  to  make  the  axis  of  the  bubble  parallel 
to  this  line. 

(a)  With  the  telescope  over  one  diagonal  pair  of  leveling- 
screws  and  the  clips  loosened,  bring  the  bubble  to  the  center  of 
its  run  ;  then  turn  the  telescope,  in  the  wyes,  a  little  to  either  side 
of  the  vertical  plane  through  the  telescope  and  note  if  the  bubble 
remains  at  the  center.  If  not,  correct  the  error  by  means  of  the 
screw  at  end  of  the  level-tube  case  arranged  for  lateral  movement. 
Repeat  until  the  tube  may  be  rotated  half  an  inch  or  more  to 
either  side  of  vertical  without  movement  of  the  bubble.  This 
adjustment  is  made  merely  to  prevent  error  from  failure  to  set 
level-tube  vertically  beneath  telescope. 

(b)  With  the  wye-clips  opened  well  out,  again  bring  the  bubble 
to  the  center  of  its  run  ;  remove  the  telescope  from  wyes  and 
turn  it  end  for  end,  then  carefully  replace  it  in  the  wyes.  Should 
the  bubble  fail  to  remain  at  the  center,  bring  it  halfway  back  by 
raising  the  lower  or  depressing  the  higher  end  of  tube  at  the 
points  of  attachment  to  telescope.     Relevel  and  repi^at  as  a  test. 


PRELIMINARY    SURVEYS. 


35 


.  49.  To  Adjust  the  Wyes  is  to  make  the  axis  of  the  telescope 
perpendicular  to  the  vertical  axis.  Witli  the  wye-clips  closed 
place  the  telescope  over  oue  pair  of  leveling-screws  aud  bring  the 
bubble  to  the  center  of  its  run  ;  then  turn  the  telescope  half-way 
rouud  on  its  vertical  axis,  so  that  its  ends  have  changed  places. 
If  there  is  any  error,  correct  by  bringing  the  bubble  half-way  back 
to  center  b}'  means  of  the  screws  connecting  wyes  with  level-bar. 
Repeat  until  the  bubble  remains  in  the  center  during  a  complete 
revolution. 

B.    Theory  of  Leveling. 

50.  Wlien  the  level  has  been  adjusted  the  line  of  collimation 
A'ill  describe  a  plane  parallel  to  the  horizontal  plane  tangent  to 
\  he  earth's  surface  at  the  point  where  the  instrument  is  placed. 
A  level  surface,  such  as  the  surface  of  still  water,  will  coincide 
with  this  plane  only  at  the  point  of  tangency,  and  will  depart 
farther  and  farther  therefrom  as  the  point  considered  recedes 
from  the  instrument.  For  short  sights  this  differeuce  may  be 
neglected  in  railroad  work,  as  will  presently  be  shown,  but  for 
long  sights  a  correction  must  be  applied. 

The  effect  of  curvature  is  to  make  objects  appear  lower  than 
they  really  are,  while  the  refraction  of  a  beam  of  light,  due  to 
the  greater  density  of  the  layers  of  air  nearest  the  earth's  surface, 
has  a  contrary  effect.  Experience  shows  the  average  error  due 
to  refraction  to  be  about  one  seventh  of  that  due  to  curvature. 


51.  The  Error  due  to  Curvature  at  any  point  is  the  deviation 
of  a  tangent  line  from  true  level,  as 
the  point  recedes  from  the  point  of 
tangency. 

Let  0  be  the  center  of  the  earth,  T 
the  point  of  tangency,  and  iVthe  point 
where  the  error  due  to  curvature  is 
desired.  Let  the  notation  be  as  shown 
in  Fig.  9.  From  the  right  triangle 
DTP,  we  have 


From  which 


Fia.  9. 


t^ 


c  = 


2R-\-  c' 

Now,   since  c  m  always  very  small  compared   with   27?,  the 
quolieut  resulting  from  the  division  of  t-  by  27?  will   not  differ 


26        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


seusibly  from  that  obtained  by  dividing  by  2R  +  c.     Therefore 
we  write 

(9) 


c  = 


2R 


For  t  =  1  mile,   M  =  3963  miles,    c  =  about  8  inches, 
for  any  other  distance  in  miles  we  have,  for  c, 

c  =  8  X  t^  inches 


Hence 


(9a) 


The    correction  for  refraction  is  about   -c,  hence  we  have, 
from  (9), 

7         7         IE 


or,  closely  enough, 


C  =  .85c. 


(10) 

Example. — "What  is  the  correction  for  a  half-mile  sight? 
For  one  eighth  of  a  mile? 

By  (9«),        c  =  8  X  ay  =  2"  for  first  case, 

and  c  =  8  X  (i)-  =  0".125  for  second  case. 

By  (10)  the  final  correction  is 

c  =  0.85  X  2  =  1".7  for  first  case, 

c  =  0.85  X  0.125  =  0.106"  for  second  case. 

52,  The  Difference  of  Elevation  between  two  points  not  so 
far  apart  but  that  a  rod  may  be  read  on  each  from  some  inter- 
mediate point  may  be  readily  found  from  these  rod-readings. 

In  Fig,  10  let  the  instrument  be  at  I,  A  and  B  the  points 
whose  difference  of  elevation  is  desired.  Let  r  =  AD,  r'  =  BC. 
Since    the    line    of    sight,  DC,    is  horizontal,  the  difference  of 


Fig,  10. 


elevation  will  evidently  be  r'  —  r.  When  the  distance  from 
/  to  A  equals  that  from  I  to  B  the  errors  due  to  curvature 
evidently  balance. 


PRELIMINARY   SURVEYS. 


27 


When  the  points  are  so  situated  that  the  rod  cannot  be  read 
on  both  from  one   intermediate   position  of  the  instrument,  an 


Fig.  11. 

auxiliary  point  or  points  must  be  used  and  readings  taken  on 
these  points  in  pairs.  Thus  in  Fig.  11  suppose  the  difference 
of  elevation  of  ^  and  5  required  : 

With  the  instrument  at  /  read  on  A  and  some  intermediate 
point  E.  Considering  the  backsights  as  plus  and  foresights  as 
minus,  the  difference  of  elevation  of  A  and  E  is  AD  —  FE. 

Again,  with  the  instrument  at  1'  the  difference  of  elevation  of  E 
and  B  is  GE  —  CB.  The  sum  of  these  differences  equals  the  dif- 
ference of  elevation  of  A  and  B,  and  may  be  written  (AB  -\-  GE) 
—  {EF-\-  CB),  or,  in  general,  the  sum  of  the  backsights  less  the  sum 
of  the  foresights  equals  tJie  difference  of  elevation. 

C.  Field-work. 

53.  A  Datum  is  a  level  surface  so  taken  that  it  shall  lie  below 
the  lowest  point  likely  to  be  reached  by  the  profile,  to  which  the 
surface  elevations  are  referred.  It  is  often  spoken  of  as  the 
datum-line  or  datum-plane,  and  is  the  zero  of  elevations. 


54.  A  Bench-mark  is  a  permanent  mark,  such  as  a  copper  or 
other  bolt  let  into  the  top  of  a  solidly  fixed  stone,  whose  height 
above  the  datum  is  known;  it  may  be  simply  a  mark  on  a  stone, 
or  a  tack  driven  into  the  projecting  root  of  a  tree,  upon  which 
the  rod  may  be  read.  In  any  case  it  must  be  so  situated  that  it 
cannot  change  its  elevation  nor  is  likely  to  be  disturbed  within 
the  lime  for  which  it  is  intended  to  be  used  as  a  standard  of 
reference. 

The  elevation  should  be  marked  on  some  object  adjacent  to 
the  bench,  with  the  letters  B.  M.  indicating  the  nature  of  the 
point. 


28        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

55.  The  Field-work  consists  in  finding  the  elevation  of  a 
number  of  points  on  the  line  established  by  transit  part}'  suffi- 
cient to  give,  when  plotted,  a  faiily  correct  outline  of  the  surface 
as  seen  in  profile. 

A  bench-mark  is  taken  at  the  beginning  of  the  line,  and  its  dis- 
tance above  mean  sea-level  or  other  datum  is  known  or  assumed. 
The  level  is  set  with  one  pair  of  leveling  screws  in  the  line  to  be 
run  (in  order  that  any  change  in  the  position  of  the  bubble  may 
be  easily  corrected;,  and  the  rod  is  read  on  the  bench.  This  read- 
ing plus  the  elevation  of  bench  gives  the  height  of  instrument 
[H.  I.)  above  the  datum. 

Readings  are  taken  at  every  hundred  feet  along  the  line,  or 
oftener  if  the  surface  changes  greatly,  until  a  point  is  reached 
beyond  which  it  is  desired  to  move  the  level.  A  peg  is  driven 
firmly  into  the  ground  and  the  rod  read  on  this  ;  the  height  of 
instrument  less  the  rod  reading  will  give  its  elevation,  as  it  will 
for  the.  intermediate  points.  This  point  is  a  temporary  bench 
and  is  called  a  turning-point.  It  should  be  marked  by  a  guard- 
stake  if  it  is  desired  to  use  it  again.  The  instrument  is  now  car- 
ried beyond  the  turning-point,  set  up,  and  the  whole  process 
repeated.  Benches  and  turning  points  should  be  read  to  hun- 
dredths or  thousandths  of  a  foot,  intermediate  points  to  tenths. 
Turning-points  are  marked  O  or  T.  P.  in  the  notes,  and  their 
positions,  as  also  the  bench-marks,  noted  by  both  leveler  and 
rodman  in  their  note-books. 


56.  The  Level  Notes  may  be  kept  in  any  convenient  form 
that  is  easily  understood.  The  following  is  used  more  exten- 
sively, perhaps,  than  any  other: 


Sta. 

B.  S. 

I 
H.  I. 

F.  S. 

Elev. 

Remarks. 

B.M. 

5.613 

205.613 

200.0 

j  B.  M.  on  root  of  L.  O.  tree  60'  to 
(     right  of  line. 

0 

2.3 

203.3 

1 

0.8 

204.8 

2 

5.7 

199.9 

3 

7.8 

197.8 

4 

99 

195.7 

O 

1.120 

196.310 

10.4-.i3 

19.5.190 

J  On  peg  at  4  -f  30'  -  20'  to  left  of  line, 
1     by  small  P.  O.  tree. 

5 

6.;i 

190.0 

6 





4.5 

191.8 

Here  the  elevation  of  tlie  datum  was  taken  200  00  feet  below 
the  tiist  bench-mark.     The  instrument  was  set  up  near  Station  2, 


PRELIMIXARY    SURVEYS.  29 

ind  a  reading  of  5.613  taken  on  the  bench;  this  was  written  in 
the  B.  S.  coluniu,  and  when  added  to  tlie  elevation  of  the  bench 
gives  the  height  of  instrument,  20o.6l3.  A  reading  of  2.3  was 
taken  on  Sta.  0,  recorded  in  the  F.iS.  column,  and  when  sub- 
tracted from  the  H.J.  jields  an  elevation  of  203.3.  The  eleva- 
tions of  other  points  were  determined  in  the  same  way.  A  little 
bej^ond  Station  4  the  rodmau  drove  a  peg  and  held  the  rod  on  it, 
yielding  a  reading  of  10.423  and  an  elevation  of  195.190.  The 
iuslrunieut  was  then  moved  to  a  point  near  Station  7  and  a  read- 
ing of  1.120  taken  on  the  peg;  this  added  to  195.190  made  the 
new  //.  /.  196.310,  and  the  process  continued  with  this  H.  I. 

In  most  cases  it  will  be  sufficient  to  read  benches  and  turning- 
points  to  hundredths  and  intermediate  points  to  tenths. 

It  will  be  seen  from  the  notes  tl»at  any  error  in -a  turning  point 
causes  the  same  error  in  all  succeeding  points.  To  guard  against 
this  the  rodman  is  required  to  keep  a  "  peg-book,"  in  which  the 
heights  of  instrument  and  elevations  of  turning-points  are  re- 
corded, and  which  must  check  with  the  leveler's  record. 

57.  Wind  and  sunshine  affect  the  accuracy  of  the  work  with 
the  level,  as  is  also  the  case  with  the  transit.  For  very  great 
accuracy  a  calm,  cloud}'^  day  is  the  best,  but  the  railroad  engineer 
cannot  always  choose  the  best  times  for  his  work,  and  must  take 
such  precautions  as  may  be  possible  while  he  exercises  the  great- 
est care  to  prevent  and  detect  errors.  The  adjustments  should 
be  tested  at  least  once  a  week,  even  when  the  greatest  care  has 
been  taken,  for  unequal  expansion  and  other  causes  may  con- 
spire to  cause  them  to  change. 

B}'  making  foresights  and  backsights  to  turning-points  about 
equal  the  error  due  to  curvature  will  be  eliminated;  the  readings 
of  rodman  at  these  points  should  also  be  checked.  The  rodman 
should  hold  his  rod  vertical,  which  is  sometimes  accomplished 
by  means  of  a  level  attached  to  rod;  or  the  leveler  can  tell  by  his 
vertical  wire  when  the  rod  is  in  the  same  vertical  plane  with  the 
instrument,  and  by  causing  the  rodman  to  wave  his  rod  back  and 
forth  slowly,  after  clamping  the  target,  he  can  tell  if  the  hori- 
zontal wire  just  bisects  the  target  at  its  highest  position. 

58.  The  Rod  should  be  graduated  to  feet  and  tenths,  reading 
by  target  at  turning-points  and  benches;  intermediate  readings 
are  maide  by  the  leveler  at  his  instrument.  Siiength  and  dura- 
bility are   essential   qualities.     The  Philadelphia  rod    seems   to 


30        A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

answer  the  purpose  as  well  as  any  other  now  manufactured;  the 
Troy  rod  may  be  used  in  the  same  manner  as  the  Philadelphia 
rod,  but  is  lighter  and  less  able  to  stand  rough  usage. 

Article  5.  The  Topographic  Party. 

59.  The  Topographic  Party  follows  the  level  and  secures  all 
the  data  necessary  for  making  an  accurate  contour-map  of  a  strip 
of  country  extending  as  far  each  side  of  the  preliminary  as  may 
be  needed  for  the  intelligent  projection  of  the  location-line. 
This  distance  may  vary  from  50  to  300  or  400  feet,  its  width  de- 
pending on  the  difficulties  to  be  encountered  and  the  degree  of 
precision  with  which  the  preliminary  approximates  to  the  final 
location-line.  The  lateral  slope  of  surface  is  obtained  at  the 
stations  of  prelimiuary  by  means  of  the  hand-level  and  tape,  by 
the  slope-level  or  clinometer,  by  cross  section  rods,  or  by  the 
transit  and  stadia.  Strictly  speaking  the  topography  includes  all 
the  surface  features,  but  for  railroad  work  the  surface  elevations, 
streams,  and  nature  of  surface  are  the  most  important;  it  may 
be  necessary  to  note  the  positions  of  roads,  buildings,  etc.,  and 
should  always  be  done  when  practicable  without  imdue  loss  of 
time.  A  pocket-compass  will  be  of  use  in  observing  the  bear- 
ings of  lines. 

60.  There  are  two  methods  of  recording  the  data  obtained; 
one  by  means  of  notes  and  sketches  in  a  book,  the  other  by 
drawing  the  contours  directly  on  the  field-sheet  as  the  data  are 
obtained.  Station  elevations  can  be  taken  direct  from  the  levelerg 
notes,  and  constitute  the  base  on  which  the  contour  elevations 
rest. 

Suppose  the  hand-level  to  be  used  and  the  notes  kept  in  a  book, 
to  be  afterwards  transferred  to  the  map.  Starting  with  the 
known  center  elevation,  the  topographer  notes  the  height  of  his 
eye  above  the  ground  and  calculates  the  height  of  center  above 
or  below  tlie  next  contour;  from  this  the  reading  of  the  rod  when 
held  on  this  contour  is  found,  being  the  height  of  station  above 
contour  {)lus  the  height  of  eye.  He  directs  the  slopeman  in  or 
out  on  a  line  at  right  angles  to  preliminary  until  this  reading  is 
given  by  the  hand-level;  the  distance  out  is  then  measured  and 
recorded,  just  as  in  setting  slope-stakes,  and  the  slopeman  di- 
rected into  position  on  the  next  contour,  in  the  same  manner. 

Thus  if  5-foot  contour-intervals  are  employed,  and  the  station 


PRELIMINARY    SURVEYS. 


31 


elevation  is  321. 6  feet  and  the  height  of  eye  5.3  feet,  we  shall  have 
for  tUe  reading  at  the  320-foot  contour  5.3  +  (321.6  -  320)=  6.9. 
Motion  the  slopeman  down  the  slope  until  his  rod  reads  6.9  and 
measure  the  distance  out,  suppose  21  feet.  Tlie  315-foot  contour 
will  be  5  feet  lower,  giving  a  reading  of  11.9,  which  may  be 
found  in  like  manner  at,  say,  80  feet  out.  As  the  rod  reads  only 
to  about  12  feet  the  topographer  must  move  out  to  this  last  point, 
and  with  the  reading  5.3+  5=  10.3  find  the  310-foot  contour  in 
the  same  way.  On  the  up-hill  side  the  325-foot  contour  will  l^e 
found  with  a  reading  of  5.3  —  (325  —  321.6)  =  1.9  feet,  and  other 
contours  in  like  manner. 
The  notes  may  be  written  thus 


Sta. 

Left. 

Center  Elev. 

Right. 

824 

305   310   315   320 
193'  125'    80 '    21 

321.6 

325   330   335   340 

27'    56'   80'  112 

The  number  above  the  line  is  the  contour  elevation,  the  num- 
ber below  its  flistance  out  from  center. 

If  preferred  the  elevation  can  be  taken  at  regular  .distances  out 
and  recorded  as  above;  the  position  of  the  contour  will  then  be 
found  by  interpolation  when  mapping  the  work. 

61.  If  the  topography  is  to  be  plotted  in  as  the  work  progresses 
the  topographer  must  have  a  light  drawing-board  with  a  pocket 
and  flap  on  back  for  holding  the  sheets  on  which  the  transit-line 
has  been  plotted  the  night  before  ;  the  station  elevations  are 
marked  on  the  line  and  the  contour  positions  spotted  in  as  ob- 
tained by  slopemen,  after  which  the  contours  are  sketched  in. 
Points  where  contours  cross  transit-line  are  found  in  the  same 
manner  as  side  points.  The  size  of  the  sheets  will  depend  on  the 
taste  of  topographer  antl  size  of  drawing-board;  17x24  to  19x28 
inches  are  good  sizes. 

The  topographer  will  soon  learn  to  guess  at  the  position  his 
contours  will  occupy  at  the  next  station  ahead,  and  will  sketch 
them  in  lightly,  to  be  erased  and  corrected  when  necessary.  It  is 
often  sufficient  to  take  lateral  readings  at  every  second  or  third 
station. 

62,  If  the  Slope-level  is  used,  the  inclination  of  the  surface  is 
obtained;  then    by  the  use  of  a  scale  constructed  to  show  the 


33        A    FJ  ELD-MANUAL    FOR   KAILROAD    EXGINEERS. 

burizontal  distance  apart  of  contours,  foi-  the  given  conlour  in- 
terval, for  slopes  varying  from  V  to  20%  the  position  of  contours 
can  at  once  be  spotted  on  the  map.  Wellington  recommends  the 
use  of  the  altazimuth  as  permitting  the  employment  of  either 
method  at  vs-ill — the  altazimuth  being  merely  a  hand-level  vfith 
a  clinometer  attached. 

63.  Cross-section  Rods  are  measuring- rods  10  or  12  feet  long 
carrying  a  level-bubble.  By  placing  one  end  at  the  center, 
bringing  the  rod  horizontal,  and  noting  the  height  of  the  end  of 
rod  on  the  downhill  side,  the  slope  may  readily  be  obtained  and 
the  contours  Avorked  in  as  before.  For  very  rough,  broken 
ground  this  method  may  be  preferable  to  either  of  the  others. 

64.  If  the  Transit  and  Stadia  are  employed,  very  elaborate 
topography  may  be  taken  with  very  little  Hekl  work,  but  the  ob 
servations  require  considerable  reduction.     AVith  a  suitable  topo 
graphic  protractor  and  the  slide-rule  mentioned  in  33,  the  large 
number  of  points  that  may  be  obtained  from  each  setting  of  the 
transit  may  be  readily  plotted  and  their  elevations  marked  on  the 
plot,  after  which  the  contour-lines  can  be  worked  in,  and  otber 
features  mapped.     For  small  vertical  angles  no  horizontal  reduc 
tion  is  needed. 

While  not  generally  favored  by  railroad  engineers  in  the  past, 
this  method  is  probably  the  most  lapid  and  economical  of  any  so 
far  employed  in  topographic  work. 

Article  6.  Preliminary  Estimates. 

65.  After  completing  the  field-work  of  the  preliminary  survey 
the  party  is  usually  disbanded,  only  the  transitman,  leveler,  and 
topographer  being  retained  to  assist  the  chief  of  party  to  complete 
the  map,  profile,  and  estimate  of  cost. 

66.  The  Map  may  be  drawn  to  any  suitable  scale,  but  less  than 
400  feet  to  the  inch  is  not  to  be  recommended  where  it  must  be 
used  in  projecting  location.  The  transit-line  is  laid  down  first 
and  the  topography  worked  in  afterwards  from  the  field-map  or 
topogiapher's  notes.  If  it  is  wanted  on  a  continuous  sheet,  the 
tiansit-line  must  first  be  drawn  on  a  succession  of  small  sheets, 
which  are  added  as  the  plotting  progresses,  a  new  sheet  being 
slipped  under  the  edge  of  the  preceding  and  tacked  tlowu  when 


PRELIMINARY    SURVEYS.  33 

required.  The  overlapping  edge  is  marked  by  a  number  of  short 
lines  extending  over  onto  the  sheet  beneath,  to  enable  one  to  re- 
place in  the  proper  position.  When  the  line  has  been  plotted  the 
sheets  are  pasted  together  and  the  whole  shifted  so  as  to  bring  the 
transit-line  over  the  continuous  sheet.  Angular  points  are  then 
pricked  through  and  the  line  drawn  on  the  continuous  sheet. 
Ordinarily  it  will  answer  to  have  the  map  drawn  on  a  succession 
of  small  sheets,  to  be  joined  together  as  required. 

The  plotting  had  best  be  done  by  bearings,  though  it  may  be 
done  from  the  delleciioii  angles,  provided  care  is  used  to  check 
frequently  by  bearings.  Otherwise  an  error  in  one  angle  wilL 
throw  all  the  remaining  portion  of  the  line  out  of  position. 

If  more  than  one  preliminar}'^  was  run,  they  should  all  be  shown 
on  the  same  sheet  whenever  possible. 

67.  The  Profile  will  be  drawn  by  the  leveler  on  profile-paper> 
and  shows  a  developed  vertical  projection  of  the  line.  The  scale 
will  depend  on  the  paper  used.  There  are  three  scales  in  general 
use,  styled  respectively  Plates  "A,"  "B,"  and  "  C."  There  is 
also  a  metric  protilo- paper.  Plate  "A"  has  the  vertical  exagger- 
ated 20  to  1  as  compaied  wilh  the  horizontal  and  is  the  best  to 
use  where  much  rock  work  is  expected.  The  vertical  exaggera- 
tion of  Plate  "  B"  is  less  than  of  Plate  "A";  this  plate  is  most 
used  for  ordinar}-  earthwork. 

A  strip  of  color  laid  on  below  the  surface-line,  and  fading  out 
at  the  lower  edge,  adds  greatly  to  the  appearance  of  the  protile. 
The  tentative  grade- line  and  points  of  change  should  be  draw-n  in 
red. 

> 

68.  Preliminary  Estimates  of  quantities  are  made  by  assuming 
a  grade-line  and  drawing  it  on  the  protile;  then  the  cuts  and  tills 
are  taken  from  the  protile,  and  the  corresponding  quantities  ob- 
tained from  Table  XIX  for  the  base  the  road  is  intended  to  have 
w'hen  completed.  The  nature  of  the  w^ork,  whether  ordinary 
earth  or  rock,  can.  of  course,  be  only  roughly  estimated. 

Bridging  is  estimated  from  the  profile  where  piling  or  framed 
bents  may  be  used,  but  where  piers  and  long  spans  are  needed 
special  surveys  with  soundings  are  required,  Culverts,  drains, 
cattle  guards,  cross-ties,  and  rails  for  main  line  and  sidings, 
switch  stands,  buildings,  right  of  way,  clearing,  and  other  factors 
entering  into  the  question  of  cost  must  all  be  considered  and 
allowed  for  in  making  up  the  estimate. 


34        A   FIELD-MAKUAL   FOR   RAILROAD   ENGINEERS. 

Engiueeriug  expenses  aud  unforeseen  outlays  that  are  sure  to 
arise  should  have  a  liberal  allowance. 

69.  The  Report  of  the  chief  of  party  should  set  forth  the  ad- 
vantages and  probable  cost  of  each  of  the  several  lines  run 
■when  there  is  more  than  one.  On  this  report  frequently  depends 
whether  or  not  the  line  is  to  be  located,  and  it  should  be  clear 
and  exhaustive,  though  plainly  and  concisely  worded.  The  map 
and  profile  form  an  integral  part  of  the  report  and  show  from 
what  data  the  estimates  were  derived. 


CHAPTER  III. 

LOCATION. 

Article  7.     Projecting  Location. 

70.  After  the  prelimiuary  lias  been  mapped  and  the  topography 
^•orked  in,  the  engineer  proceeds  to  make  a  paper  location  for  his 
guidance  in  the  field.  The  solution  of  the  varied  and  complex 
problems  that  confront  him  are  more  or  less  interdependent. 
The  guiding  principle,  applicable  to  all  departments  of  engineer- 
ing, that  the  best  structure  is  that  which  for  the  least  cost  test  an- 
swers the  purpose  for  uliich  iticas  intended,  should  control,  even 
though  the  resulting  structure  be  inferior,  in  point  of  scientific 
design,  to  some  other.  The  best  road  as  regards  construction  and 
grades  may  be  a  failure  because  of  excessive  first  cost,  while 
the  cheapest  construction  will  entail  such  heavy  operating  ex- 
penses that  it  may  be  equally  unprofitable.  The  alignment  must 
be  as  free  from  curves  as  possible,  while  heavy  grades  are  at  the 
same  time  excluded;  these  two  requirements  conflict  and  must 
be  as  well  adjusted  as  possible.  The  amount  of  earthwork,  of 
bridging  and  other  structures  must  be  kept  down  to  the  lowest 
limits. 

71.  Starting  at  the  summit  of  the  most  diflScult  portion  of  the 
route,  assume  a  starting-point  and  elevation;  with  the  dividers  set 
at  such  a  distance  to  the  scale  of  the  map  as  will  give  a  fall  of  one 
contour  space — or  half  space — for  the  assumed  grade,  step  down 
the  slope  in  such  a  way  that  the  dividers  fall  each  lime  on  the 
next  lower  contour,  oi  half-space,  according  to  the  fall  assumed  in 
setting  dividers.  If  curve  compensation  is  allowed,  the  dividers 
must  be  reset  for  each  curve,  for  the  same  fall,  since  the  giade 
will  be  slackened  on  curves.  The  points  at  which  the  dividers 
fall  are  lighily  ..potted  on  the  map  and  connected  b}'^  a  grade 
contour,  which  represents  the  surface-line  having  tlie  required 
gradient.     This  line  will  be  too  broken  to  be  used  as  a  ]or"tiou- 

35 


36        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

Hue,  so  we  have  then  to  draw  on  the  map  a  succession  of  curves 
and  tangents  that  will  approximate  sufficiently  close  to  it,  at  the 
same  time  that  a  proper  balance  is  maintained  between  earthwork 
and  curvature. 

Having  lightly  plotted  the  proposed  line,  the  elevations  are 
transferred  to  profile-paper,  thus  giving  a  profile  of  the  line. 
With  a  fine  thread  stretched  aloug  the  profile,  to  represent  the 
grade-line,  adjust  the  cuts  and  fills  to  suit  the  nature  of  the  work, 
la  general,  fills  are  cheaper  than  cuts  both  in  construction  and 
maintenance;  and  especially  is  this  true  Avhere  a  shallow  surface 
layer  of  earth  is  underlaid  by  rock.  It  may  happen  that  the 
material  from  excavation  must  be  used  in  embankment,  when 
the  cuts  and  fills  must  be  made  to  balance  b}'  shifting  the  grade- 
line  until  this  appears  to  be  the  case  on  the  profile. 

At  the  stream  crossings  the  grade-line  must  be  kept  safely 
above  high- water  mark,  so  that  sufficient  waterway  is  provided, 
and  allowance  made  therefor. 

After  locating  the  most  difficult  portions  pass  on  to  the  easier 
work,  returning  later  on  .to  study  the  effect  this  will  have  on  the 
part  first  located.  It  may  be  necessary  to  go  over  the  projection 
several  limes  before  you  can  be  reasonably  sure  that  the  best  loca- 
tion has  been  projected;  even  then  the  study  of  tlie  line  in  the 
field  will  cause  many  of  the  details  to  be  altered,  sometimes 
materially. 

Long  grades  are  to  be  preferred  to  short  ones,  but  questions  of 
economy  may  necessitate  the  latter  in  order  to  lighten  work;  care 
must  be  taken  that  the  grades  are  not  so  badly  "  chopped"  that 
they  interfere  with  the  easy  riding  of  the  train. 

In  projecting  the  line  it  will  generally  be  best  to  strike  the 
curves  first  and  draw  the  tangents  afterwards,  though  it  some- 
times happens  that  long  tangents  will  control  the  curves;  when 
this  is  the  case  the  tangents  are  drawn  to  intersection  and  the 
curves  afterwards  put  in. 

When  transition-curves  are  employed,  a  slight  offset  should  be 
made  at  the  beginning  and  end  of  curves  to  allow  for  their  inser- 
tion in  the  field.  These  offsets  will  be  so  small  that  it  is  useless 
to  attempt  to  show  them  to  scale. 

72.  A  Curve-protractor  will  be  of  material  assistance  in  find- 
ing the  degree  of  curve  required  to  unite  two  tangents  that  have 
been  laid  down  on  the  map.  It  consists  of  a  transparent,  semi- 
circular protractor  having  a  series  of  curves   from  30'  up  to  8° 


LOCATION.  37 

plainly  cut  upon  it.  The  curves  are  on  both  sides,  those  on  the 
reverse  side  having  their  concavities  turned  in  an  opposite  sense 
from  those  on  the  face.  The  scale  is  usually  400  feet  to  the  inch, 
and  in  any  case  the  map  and  protractor  must  be  drawn  to  the 
same  scale.  Sometimes  a  set  of  cardboard  or  hard-rubber  curves 
are  used,  but  they  are  inferior  to  the  curve-protractor.     To  use 

,  it,  simply  prolong  tangents  to  intersection  and  then  place  the 
protractor  so  thai  the  curve  admitting  of  tlie  best  grade  is  tan- 
gent to  the  two  straight  lines.  Mark  the  points  of  taugency, 
which  will  be  the  beginning  and  end  of  curve.  When  the  curve 
is  required  to  pass  through  a  given  point  the  proper  curve  may 
be  immediately  found  by  trial,  whereas  the  calculations  would 
require  some  little  time. 

^-«7j*^  Reversed  curves  should  never  be  allowed  on  main  lines.  Suffi- 
cient tangent  should  be  interposed  to  allow  space  for  easing  off 
the  superelevation  of  outside  lails,  or  for  the  insertion  of  tran- 
sition-curves when  these  are  w  be  employed. 

73.  The  Field  Corps  is  substantially  that  required  on  the  pre- 
liminary survey,  and  the  methods  of  work  pretty  much  the  same, 
except  that  curves  must  now  be  run  in,  and  this  necessitates  more 
clearing.  If  first  and  second  location-lines  are  to  be  run  (and  it  is 
real  economy  to  run  both),  it  will  not  be  necessary  to  have  the 
stationing  continuous  on  the  lirst,  so  the  pluses  arising  from 
*'  backing  up"  need  only  be  noted  and  eliminated  when  the  final 
location-line  is  run.  If  transition-curves  are  to  be  inserted,  they 
need  not  be  run  the  first  tiixl^,  the  proper  offset  being  made  at 
vhe  P.  T.  or  P.C  of  the  circular  curves,  which  latter  are  to  be  run. 

On  the  final  location-line  the  stationing  must  be  continuous, 
beginning  with  zero.  The  stakes  are  marked  as  on  the  pre- 
liminary survey,  and  all  hubs  that  are  likely  to  be  used  again 
must  be  referenced  in,  the  reference-hubs  being  set  well  out  of 
the  way  of  disturbance  by  the  plow  or  scraper. 

The  leveler  should  make  bench-marks  every  1000  or  2000  feet, 
to  be  used  in  running  check-levels  and  in  giving  grades  later  on. 

From  the  paper  location  the  notes  should  be  made  up  in  the 
office,  to  serve  as  a  guide  in  the  field;  however,  no  attempt  should 
be  made  to  adhere  rigidly  to  them,  since  slight  errors  in  the 
mapping  will  affect  the  projected  line,  while  in  the  field  the  line 
may  be  shifted  here  and  there  so  as  to  fit  the  ground  more  snugly 
and  accord  more  closely'  with  what  the  nature  of  the  earthwork 
demands. 


38        A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

The  highest  skill  of  the  engineer  is  required  to  secure  the  best 
location-line,  and  he  should  have  all  the  time  he  needs.  Undue 
haste  on  location — as  on  reconnoissance  and  preliminary — is 
almost  sure  to  result  in  increased  cost  of  construction. 


Article  8.    Simple  Curves. 


4.  Definitions  and  Formulas. 

74.  The  Circular  Curves  that  are  usually  employed  to  unite 
straight  reaches  of  the  railroad  may  be  simple,  compound,  or  re- 
versed. The  use  of  reversed  curves  should,  however,  be  limited 
to  turnouts  and  cross-overs. 

a.  A  Simple  Curve  is  the  arc  of  a  circle. 

b.  A  Compound  Curve  consists  of  two  simple  curves,  of  differ- 
ent radii,  both  on  the  same  side  of  a  common  tangent. 

c.  A  Reversed  Curve  is  made  up  of  two  curves  of  contrary 
flexure  having  the  same  or  different  radii,  and  a  common  tangent. 

d.  The  Point  of  Curve  {P.O.)  is  the  end  of  tangent  and  begin- 
ning of  curve,  as  at  A,  Fig.  13. 


Fig,  12, 


e.  The  Point  of  Tangent  {P.T.)  is  the  end  of  curve  and  be- 
ginning of  tangent,  as  at  B  of  Fig.  12. 

/.  The  Point  of  Intersection  (P.I.)  is  the  point  where  the 
tangent  at  theP.C.  and  P.T.  intersect  when  produced.  {D  of 
Fig.  12.) 

g.  The  Intersection  Angle  (7^  is  the  angle  at  the  PL  be- 
tween the  tangents  meeting  there,  and  equals  the  angle  at  the 
center. 

h.  The  Tangent  Distance  (7")  is  the  length  of  the  produced 
tangent  measured  from   the /*.  6'.  ov  P.T.  to  the  P./,     The  term 


LOCATION. 


39 


tangent  is  applied  to  aii}'^  straight  portion  of  the  Hue,  but  the  letter 
T  will  be  used  to  designate  the  produced  portion  only. 

i.  The  Mid-ordinate  {M)  is  the  portion  of  the  radius  inter- 
cepted between  the  arc  and  chord  when  it  cuts  the  chord  at  its 
middle  point. 

j.  The  External  {E)  is  the  part  of  the  radius  produced  to  the 
P. I.,  intercepted  between  curve  and  the  P.l. 

k.  The  Long  Chord  (L.C.)  is  the  chord  joining  the  P.O.  and 
P.T.  Frequently  the  term  is  applied  to  any  chord  longer  than 
the  unit  chord. 

l.  The  Radius  will  be  denoted  by  R. 

tn.  The  Point  of  Compound  Curve  (P.  G.  C. )  is  the  point  of 
common  tangency  of  the  two  branches  of  a  compound  curve. 
(See  Fig.  13.) 


P.C. 


n.  The  Point  of  Reversed  Curve  {P. B.C.)  is  the  point  of 
common  tangency  of  the  two  branches  of  a  reversed  curve. 

o.  The  Degree  of  Curve  {P)  is  the  angle  at  the  center  sub- 
tended by  the  unit  chord.  In  the  United  States  this  chord  is  100 
feet,  in  England  66  feet,  and  where  the  metric  system  is  em- 
ployed it  is  taken  at  20  meters.  Any  convenient  chord  length 
may  be  taken,  but  for  uniformity  American  engineers  have 
adopted  the  chord  of  100  feet,  and  unless  otherwise  stated  it  is 
always  so  understood  when  we  speak  of  the  degree  of  curve. 

Half  the  degree  of  curve  is  called  the  deflection-angle,  since 
it  is  the  angle  to  be  deflected  from  the  tangent  to  the  chord. 

If  there  were  any  practical  method  ot  measuring  around  the 
curve  instead  of  along  the  chord,  an  accurate  and  convenient 
ratio  for  expressing  the  radius  in  terms  of  the  degree  would  be 
had.  Thus  if  D  is  the  angle  at  the  center  subtended  by  the  mre 
of  unit  length,  we  have,  where  a  is  this  uuit  arc, 


40        A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


Hence 


27tR=z  a    -jj. 


„       a     360 


•  •  c 


(11) 


When  a  equals  100  ft,  this  becomes 

100    360 


R  = 


27t      D 


(11') 


i2  varies  inversely  as  B,  so  that  knowing  the  radius  for  a  1° 
curve,  we  should  have  only  to  divide  this  by  D  to  get  the  radius 
for  a  D^  curve. 

Since  the  chord  is  employed  instead  of  the  arc,  we  determine 
R  by  means  of  the  following  problem 

75.  Given  the  Chord  C,  and  Degree  of  Curve  D,  to  Find  the 
Radius  R. 

In  Fig.  14,  AB  \s  the  chord  G,  OE  a  perpendicular  from  the 
center  upon  AB 


From  the  right  triangle  AEO  we  have 
R  sin  IB  =  10. 


Whence  R  = 

When  CislOOft., 


.     ,  „  =  iCcosec  IB. 

sm  IB       "  ^ 


.    .     (12) 


50 
R  =  —. — r-^  =  50  cosec  IB. 
sni  IB 


.     .     .     (12') 


LOCATION".  41 

Comparing  results  givcu  by  formula  (13')  with  those  given  by 
(11'),  we  have  for  a  few  curves: 

Degree  of  Curve.  R  by  (12').  R  by  (11').          Difference. 

1 5729.65  5729.58  0.07 

2 2864.93  2864.79  0.14 

3 1910.08  1909.86  0.22 

5 1146.28  1145.92  0.36 

7 819.02  818.51  0.51 

10 573.69  572.96  0.73 

14 410.28  409.26  1.02 

20 287.94  286.48  1.46 

The  difference  is  seeu  to  be  about  one  half  a  foot  for  a  7° 
curve,  one  foot  for  a  14°  curve,  and  one  and  one-half  feet  for  a 
20°  curve. 

Up  to  a  7°  curve  the  difference  is  inconsiderable,  and  we  may 
stake  out  curves  with  100- foot  chords.  From  7  to  14  degrees  50- 
foot  chords  may  be  used.     Therefore 

25 

R  —  - — r^  =  25  cosec  \D (12'a) 

sin  \D  * 

For  curves  from  14°  to  28°  we  should  use  25-foot  chords, 
for  which 

12  K 

R  =  ^-—f,  =  12.5  cosec  ^D {12'b) 

sin  ^D  " 

Above  28°  shorter  chords— say  10  feet— should  be  used,  if  the 
curve  cannot  be  struck  from  the  center.     In  this  case 

R  =  - — --^  =  5  cosec  4^B (12'c) 

sin  ■^\I) 

Table  I  of  radii  was  computed  by  formulas  (12'),  (12'a),  and 
(12'6). 

In  practice  it  is  customary  to  take  the  radius  of  a  1°  curve  as 
5730  feet  and  to  assume  the  radii  to  vary  inversely  as  the  degree ; 
thus  for  a  4°  curve  the  radius  wouhl  be  R  =  ^p^  =  1432.5  feet, 
while  by  Table  I  it  is  1432.69  feet — a  difference  of  only  .19  foot ; 
for  a  12°  curve  R  =  m^  =  477.5  feet,  wliile  by  Table  I  it  is 
477.68  feet.  The  effect  of  taking  5730  instead  of  5729.65  for  the 
radius  of  a  1°  curve  is  to  reduce  the  error  resulting  from  the 
assumption  that  U  equals  5730  divided  by  the  degree  of  curve. 


42        A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

76.  The  Length  of  Curve  (L)  is  foiiud  by  dividing  the  angle 
at  the  center  (which  equals  the  intersecliou  angle)  b}'  the  degiee 
of  curve,  the  result  being  in  chains  and  decimals  of  a  chain.  The 
number  of  P.O.  -\-  L  will  give  the  station  number  of  P. T. 

Example.— The  P.  C.  of  a  4°  curve  having  /=  26  30'  is  at  sta. 
104  +  12.5.     Find  L  and  the  number  of  the  P. T.     Here 


Of!    K 

X  =  '^  =  6.625  chains. 


104.125  +  6.625  =  110.75  ;  hence  the  number  of  P.  T.  is 
110  +  75. 

77.  Use  of  the  Table  of  Functions  of  a  One-degree  Curve. — 

In  the  location  of  railway  curves  geometrical  accuracy  will 
frequently  be  of  less  importance  than  rapidity  of  Held- work,  so 
long  as  errors  are  kept  within  certain  limits. 

On  tangents  slight  errors  of  alignment  may  readily  be  detected 
by  the  unaided  eye,  but  on  curves  these  are  not  .so  apparent. 
Moreover  it  is  not  likely  that  the  trackmen  will  keep  them  up  in 
the  exact  position  of  their  location. 

To  simplify  and  shorten  the  field  computations  engineers  make 
use  of  a  table  of  functions  of  a  1°  curve,  and  assume  these  func- 
tions for  other  curves  to  vary  inversely  as  their  degree,  or  directly 
as  their  radii.  Table  IX  gives  values  of  the  tangent  distances, 
long  chords,  mid-ordiuates,  and  externals  for  a  1°  curve,  the 
radius  of  which  is  taken  as  5730  feet.  To  find  these  functions 
for  other  curves,  divide  the  tabular  values  by  the  degree  of  curve. 
The  error  resulting  from  this  assumption  will,  in  any  practical 
case,  amount  to  no  more  than  a  few  tenths  or  hundredths  of  a 
foot. 

Table  IX  may  also  be  used  as  a  metric  curve  table,  the  tabular 
values  being  taken  as  meters  instead  of  feet.  If  the  unit  metric 
chord  is  20  meters  long,  this  may  be  taken  as  one  fifth  of  the 
tabular  unit  chord;  so  to  use  the  table  multiply  the  metric  degree 
by  5  and  enter  the  table  with  the  result  as  a  value  of  D. 

For  instance,  a  2°  metric  curve  having  /  =  40°  w^ould  have  a 

mid-ordinate  equal  to  ,r— — .:  =  34.56  meters. 
^  2X0 

For  the  approximate  radius  of  a  metric  curve  divide  5730  by  5 

times  the  degree.     Thus  a  4°  metric  curve  would  have  R= 

4X5 


LOCATION".  43 

—  286.5  meters.     For  the  exact  radius  make  use  of  formula  (12). 

Thus  for  a  4°  curve  haviuff  20-mcter  chords  K  =  — — ^  =  286.54 

°  sm  2 

meters,  a  difference  of  only  .04  meters. 

If  a  metric  curve  is  to  be  retraced  with  a  100-ft.  chain,  we 
convert  the  metric  degree  to  the  degree  referred  to  100-ft.  chords 
by  the  relation  that  a  100-ft.  chain  =  1.524  chains  of  20  meters 
each;  a  20-meter  chain  =  65.618  ft.;  one  foot  =  0.8048  meters; 
one  meter  =  3.2809  ft. 

It  will  sometimes  be  a  sufficiently  close  approximation  to  take 
the  20  meter  chain  as  two  thirds  of  a  100-ft.  chain;  this  will  make 
the  metric  curve  nearly  two  thirds  of  the  degree  the  same  curve 
would  have  when  laid  out  with  a  100-ft.  chain,  and  the  curve  with 
100-ft.  chords  nearly  three  halves  of  the  degree  as  laid  out  with 
the  20-meter  chain.  Thus  a  4°  metric  curve  would  be  equivalent 
^o  a  6°  curve  laid  out  with  a  100-ft.  chain. 

In  the  problems  that  follow  two  methods  of  solution  will  be 
given  when  practicable — the  first  being  rigid,  while  the  second 
.'s  based  ou  the  use  of  Table  IX.  To  shorten  the  formulas  the 
subscript  1  will  be  written  after  the  letters  T,  L.G.,  M,  and  E 
when  these  are  the  functious  of  a  1°  curve.  Thus  Ti  '4-  28"  means 
Mie  tangent  distance  for  a  1°  curve  when  /=28°,  L.G.i  '4-  16° 
Mie  long  chord  for  a  1°  curve  when  /=  16°,  etc. 

78.  Tables  of  Natural  and  Logarithmic  Circular  Functions.  — 
Many  engineers  prefer  to  work  altogether  by  tables  of  natural 
sines,  cosines,  etc.,  and  time  may  often  be  saved  by  their  use. 
Nevertheless  logarithmic  tables  are  of  frequent  advantage,  even  in 
the  field,  and  the  more  important  ones,  such  as  the  logarithmic 
sines,  cosines,  tangents,  and  cotangents,  together  with  the  loga- 
rithms of  numbers,  are  given  in  the  back  of  the  book  along  with 
the  tables  of  natural  functions. 

79.  Given  R  and  G  to  Find  D. 
From  equation  (12), 

smlD  =  i^ (13) 

JX 

80.  Given  /  and  II  (or  D)  to  Find  T. 

If  I)  is  given,  find  li  by  (12);  then  in  Fig.  15  from  triangle 
OAB  we  get 

T=2iiiinU. (14) 


44        A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

By  Table  IX. — Fiud  the  tabular  value  of  T  for  the   given 
angle  I;  then 


T  = 


D 


(14a) 


Example.— /=  35°  40',  Z)  =  4";  required  T. 
By  (14),      T=  1432.69  tan  17°  50'  =  460.91  feet. 

By  (14a),  T  =  — -^-  =  460.85  feet,  a  result  differing  from  the 

value  found  by  tbe  rigid  method  by  ouly  0.06  foot. 


81.  Given  I  and  Tto  Find  R  or  D 

From  (14), 

T 


R  = 


tan  \I 


=  TcotU. 


Then  by  Table  I  the  degree  may  be  found. 
By  Table  IX. 


D  = 


(15) 


(15a) 


82.   Given  /and  D  to  Find  the  Long  Chord  L.G. 
First   find  R  by  (12)  or  (12'),  or  by  Table  I  ;  then  from  the 
triangle  O^Fof  Fig.  15, 


AF=R?>m  II. 

AG  =  2AF=  L.a  =2R  sin  ^I.    . 


(16) 


LOCATION. 


45 


By  Table  IX. — Fiud  the  tabular  L.C.  for  the  giveu  augle  /; 
tbeu 


L.G.= 


Xy.  C.J 


(16a) 


83.  Given  the  Radius  R  and  any  Chord  G  to  Find  the 
Ordinate  to  the  Curve  at  any  Point. 

First  Method.— In  Fig.  16  let  HE  be  the  chord  C\  HK  =  a 
and  KE  =  b,  the  segments  into  which  it  is  divided  by  the  ordi- 


nate y.     Draw  the  radius  through  K;  call  the  portion  between 
chord  aud  curve  y'.     By  geometry, 


from  which 


(2R  -  y')y'  =  ah, 

,  ah 

y 


^n-y" 

But  y'  is  small  compared  with  27?,  and  hence  we  write 

ab 


^=2li 


(a) 


Now  y  does  not  differ  sensibl}^  from  y'  in  the  cases  met  with  in 
practice,  so  we  write 


y  = 


ah 
2M 


(6) 


46        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

If  we  write  E  =         -,  formula  {b)  becomes 

_      abD 
^  ~  a  X  5730 ^^^ 

a  b 

Let  — -  =  m     J—  =  n,  and  substitute  in  (c),  giving 

y  =  TTTgQ  mnD  =  O.SldmnD, 

or  very  nearly 

y  =  |wnZ> (17) 

y  is  given  in  feet  when  m  and  n  are  in  chains  and  decimals  of 
a  chain. 
At  the  mid-point  F,  m  =  n,  and  y  =  M. 

.-.31=  pi^D (18) 

Caution. — Formulas  (17)  and  {\S),  while  ver}'  convenient  for 
field  use  in  passing  obstructions,  are  liable  to  error  when  very 
long  chords  or  large  values  of  I)  are  used,  since  Ihey  give  results 
that  are  too  small. 

If  we  write  the  arcs  HN,  NE  for  a  and  b,  we  shall  get  results 
that  are  too  large,  3'et  about  Jis  near  the  true  values  as  by  taking 
m  and  n  to  be  the  segments  of  the  chord.  To  illustrate  we  will 
find  a  few  values  of  J/  and  compare  with  the  true  values  taken 
from  Table  V. 

Decree  Length  Mid-ord.  Mid-ord.  Mid-ord. 

of  of  by  by  by 

Curve.  Arc.  M^i{HF)-^D.      M=l{HGyW.     Table  V. 

2 2  stations.  1.75  1.75  1.75 

2 G  "  15.69  15.75  15.69 

5 2  "  4.37  4.38  4.36 

5 6  "  38.51  39.38  39.06 

8 2  "  6.96  7.00  6.97 

8 4  "  27.29  28.00  27.75 

8 5  "  42.02  43.75  43.20 

8 6  "  59.43  63.00  61.93 

From  this  it  appears  we  may  use  formula  (18) — and  (17)  as 
well — taking  eitlici-  llie  segments  of  the  arc  or  chord  for  curve.'! 
not  exceeding  4°  with  arcs  up  to  600  ft.;  for  curves  from  4°  to  6" 


LOCATION.  47 

tbej  may  be  used  up  to  500-ft-  arcs,  while  for  curve§  Ijetwedu 
6°  !ind  8'  uot  more  Ibrm  400  feet  of  arc  may  be  takeu. 

SF.co^'D    Method— First    determine    the    mid-ordinate.      In 
triangle  OEF, 


0F=  \^W  -  \(T-\ 
then 


M=FG  =  R-  VR'  -  \G' (19) 

To  find  ordinate  J.C  distant  d  from  the  mid-point  ot  EH,  draw 
OB=d  parallel  to  HE;  draw  AB  at  right  angles  to  HE.     Then 


BA  =  x^R' 
Therefore 


CA  =  y=  VR'  -  d^  ->  \/R'  -  \0.    .     .     .    (20) 

Third  Method. — If  the  chord  C  is  short,  we  mav  resrard  the 
arc  as  an  arc  of  a  parabola,  for  which  it  is  kuo-A-n  that  ordi- 
nal es  vary  as  the  product  of  the  segments  into  which  they  divide 
the  chord.     The  mid  ordinate  being  known,  we  have 


.•.y=4«-^. (21) 

From  formula  (b)  we  have  for  y  =  M,  a  =  b  =  ^C, 

The  mid-ordinate  for  any  other  chord  C"  is 

Hence 

M~  C 

.-.  M,  =  m{^J (23) 

If  C  =  ^C,  this  gives 

M,=iJtr. (23) 


48        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

This  last  relation  affords  an  easy  method  of  staking  out  a  curve 
when  the  mid-ordiuate  of  a  given  chord  has  been  determined. 
First  erect  the  ordinate  iLT  at  the  midpoint  of  the  chord;  then 
join  the  ends  of  chord  with  the  extremity  of  the  ordinate  just 
measured;  the  lengths  of  these  chords  do  not  differ  much  from 
IC;  at  their  mid-points  erect  ordinates  equal  to  ^M,  giving  points 
on  the  curve.  Proceed  in  like  manner  for  other  points  until  a 
sufficient  number  have  been  located. 


84.  Given  i?  and  /  to  Find  the  External  E. 
In  Fig.  n  E=GB=OB-  OG. 
But  OB=R  sec  \I  and  OG  =  R. 

.  •.  E=.R{sec  \l-\)  =  R  ex  sec  i/.     .     .     .      (24) 

By  Table  IX. — Find  E  for  a  1°  curve  for  an   intersection 
angle  I;  then 


E  = 


D 


(24a) 


85.  Given  T  and  /  to  Find  E. 

In  Fig.  17  draw  BG  perpendicular  to  AB,  and  produce  AG  \,o 

b/ 


intersect  5Cat  C.  BG\%  parallel  to  AG,  and  the  triangles  AGO 
and  GBCfiva  similar;  hence  BC  =  BG  =  E.  In  the  right  triangle 
ABC,  angle  BAG=\BAF=  \I.     Therefore 


^  =  r  tan  \L 


(25) 


Exercise.  — Derive  equation  (25)  from  (24). 


LOCATION. 


49 


86.   Given  3/ and  /to  Find  E. 
FroDi  trigonometry, 

IT  1 

sec  4/  =  r>. 

'  COS  \I 

Insert  this  in  (24)  and  we  get 

.1  —  cos  47 


E=R 


cos  |/ 


(a) 


But  from  Fig.  17,  M  =  R{\  -  cos  \I),     Substitute  in  {a) : 

M 


E  = 


cos  \I 


=  M  sec  4/. 


(26) 


87.  Given  £^and /to  Find  i?. 

From  (24), 

E  E 


R  = 


sec  |/  —  1       ex  sec  ^/ 

88.  Given  /and  ^ to  Find  /. 
From  (25), 


_,  cos  4/ 

=  E 5_ 

vers  |/ 


*  s 


(27) 


(28 


89.  Given  the  Chord  C  and  Degree  of  Curve  D  to  Find 
the  Chord  Deflection  Offset  d. 
In  Fig.  18  extend  EA  to  H,  making  AH  =  EA  =  AB;  join 


"O 
Fig.  18. 
U  and  5  and  draw  AK  to  the  mid-point  of  HB.    Then 

/Zir  =  KB=  C  sin  i/). 
.-.  d=  IJB  =  20  sin  W 


(29) 


50        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEEHS. 

When  C  =  100', 

d  =  200  sin  ID (29') 

If  we  write  sin  1^  =  ^  from  (12)  in  formula  (29),  there  results 

^  =  -^ (30) 

For  curves  up  to  7^  C  =  100';  hence 

^   10000 

^=-E- (3^') 

For  curves  from  7"  to  U\  C  =  50';  therefore 

<^  =  f» (30") 

5730 
For  R  write  —  ,  and  (30'),  for  C  =  100,  becomes 

-i:?-----^ <-, 

and  for  C  =  50.  (30")  becomes 

2^00  7) 

^-5-:^^  =  .43632)  =  . 873.-.     .     .     .      (31') 

Example.— Find  d  for  a  6°  curve,  C  =  100  feet. 
By  (29'),  cf  =  200  X  0.05234  =  10.47  feet. 

By  (30'),  d  =  ]^^  =10.47  feet. 

''        '  955.4 

By  (31),  d  =  1.745  X  6  =  10.47  feet. 

90.  Given  the  Chord  G  and  Degree  of  Curve  D  to  Find  the 
Tangential  Deflection  Offset  t. 

lu  Fig.  18  make  EF  (tangent  at  E)  equal  to  EA,  and  join  F 
with  A.  Draw  EG  to  the  mid-point  of  FA.  Angle  AEG  = 
GEF  =  \D;  hence,  from  the  figure, 


AG  =  GF=  Csm^D. 

:  t  =  2C  sin  ID (32) 


LOCATION. 


51 


When  C  =  iOO  feet, 

t  =  200  siu  \D. 


(33) 


Since    \D   is  small,   we   may   write,    without   material   error, 
siu  ID  =:  ^  sin  |Z);  then,  writing  siu  W  =  -n,  as  in  89,  we  get 


t  = 


91 

211 


5730 


Making  C  =  100  ft.  and  writing  11  —  — ^  gives 


i  = 


10000 


1) 


D  =  0.873i>. 


2  X  5730 
When  C  =  50  feet,  (33)  yields 

t  =  .218Z>  =  .436  X 


D 


(33) 


(33') 


(33") 


Example.— Fiud  t  for  a  6°  curve,  G  =  100  ft. 
By  r32')  t  =  200  sin  1°  30'  =  5.24  ft. 

By  (33'),  =  .873  X  6  =  5.24  ft. 

91.  To  Find  the  Subtangential  Deflection  Offset  i'  for  a 
Subchord  C" 

First  Method. — By  formula  (13)  fiud  the  angle  at  the  center 
subtended  by  the  subchord  C;  call  this  angle  D'.     From  (32), 

i'  =  2C'  sin^D'.  . (34) 

Second  Method.— In  Fig.  19,  with  ^as  center  strike  the  arcs 
FG  and  AH,  taking  EF  =  C  and 
EA=  C;  prolong  EG  to  B.  Now 
assuming  that  the  chords  C  and  G 
are  proportional  to  their  central 
angles  we  have 


From  the  similar   sectors  EFO 
and  EAB,  since  EB  =  G, 


Fia,  19. 


AB 


C' 

t'  ' 


(*) 


^o 


>'Z        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


MiiltiplyiDg  (a)  and  (b)  together,  term  by  term, 


Whence 


C_l      C^ 
C'~  t'  '  C 


'-{?)■ 


(35) 


Example. — Find  t'  for  a  7°  curve  when  (7  =  60  ft. 


Here 

ly 

By  (34), 

t' 

By  (32'), 

t 

By  (35), 

t' 

=  ^  X  7°  (very  nearly)  =  4'  12'. 

=  2  X  60  X  0.01832  =  2.20  ft. 
=  6.11  ft. 


92.  To  Find  the  Tangent  Offset  z. 

In  Fig.  20,  EB—z  is  the  required  offset.    Let  AE=  n  chains  = 

lOOn  feet.  AE=FB,  the  half-chord 
having  the  mid-ordinate  AF  =  EB , 
hence  we  have,  by  formula  (18), 

2  =  In-I).     .     .     .     (36) 

In  this  formula  we  may  take  n  to 
he  either  the  length  of  AE  ov  the  arc 
AB,  in  chains.  If  taken  equal  to  AE 
the  offsets  will  be  slightly  too  small, 
while  if  taken  equal  to  AB  they  will 
be  a  little  too  large.  The  use  of  the 
formula  is  limited  to  small  values  of 
n  and  B,  as  was  pointed  out  in  83. 
(See  Caution.) 
Formula  (36)  is  easy  of  application  and  of   frequent   use  in 

locating  curves  by  offsets  from  the  tangents.     For  curves  up  to 

4°   71   may  be  as  great    as  3,  but  for  sharper  curves  it  should 

be  less. 

Example. — Find  six  offsets  to  a  4°  curve  at  points  50  ft.  apart, 

measured  around  the  curve. 


Fig.  20. 


LOCATION. 


53 


By  successive  applicatious  of  (36)  we  have 


for  71 
n 
n 
n 
n 
n 


1 


=   1,       2  = 


—  3 

—  2' 

—  9 


5 


=  |X    i  X4  = 


=  3,     2  = 


I  X  1   X  4 

i  X  I  X  4 

i  X  4x4 

I  X  -f  -  X  4 

i  X  9X4 


0.88  feet 

3  50 

7.88 
14.00 
21.88 
31.50 


The  last  vahie  of  z  is  iu  error  by  about  0.2  ft  ,  but  for  seltiug 
stakes  on  coii.structiou  this  difference  is  not  material  so  long  as 
the  alignment  beyond  this  point  does  not  depend  on  it.  In 
setting  irack-cenlers  the  completed  road-bed  is  available  and  the 
stakes  may  be  set  with  the  transit,  in  the  usual  way. 

93.  Diflference  in  Length  of  a  Circular  Arc  and   its   Long 
Chord. 
First  Method. — Let  the  central  angle  be  a  degrees.     By  (13), 
c 


sin  \a°  ■= 


^R 


Changing  degrees  to  circular  measure,  a  (in  n  meas.)  = 

:  — — .     The  length  of  arc  is  Ra  =  R^ir^-     Then 
57. o  57. o 


It  a 
180 


a 


Arc  —  chord  =  Rzrrr^  —  ^■ 

5<.o 


(37) 


Second  Method. — An  easy  approximation  may  be  found  as 
follows : 

Referring  to  Fig.  17,  ^^=c,  QF=M.     luQi  AG  =  b  =  ^-\-x. 

From  the  right  triangle  AFG 


'i+-y=i+^'- 


From  which 


X 


c  -{-  x 


(a) 


54        A   FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

Neglectiug  the  x  in  deuomiuator  as  small  compared  with  c 
gives 

^=T (*> 

Then  will  26  -  c  =  2^  =  — (38) 

From  Huygens'  approximation  to  the  length  of  a  circular  arc 

(see   Williamsou's  Differential   Calculus,  p.  66),  arc  =  — ^ — . 

o 

Therefore 

Arc  —  chord  =  — c  =  |(25  —  c).      .     .       (c) 

o 

Inserting  the  value  of  2o>  —  c  from  (38)  gives 

8i/'- 
Arc  —  chord  =  —^ — {d) 

When  the  arc  is  not  very  great  we  may  write  c  =  100/h  ,  where 
7ii  is  the  number  of  chains  contained  in  the  arc  AE.  From  (18), 
remembering  that  rii  =  2n, 

M  =  0.21Snr'J). 
Inserting  these  values  of  c  and  M\n  (d), 

Arc  -  chord  =  |  ^^^^""-^  =  ^^n,WK  nearly.  .     (39) 

Example. — Find  the  difference  in  length  of  arc  and  chord  of 
a  4'  curve  when  ni  =  6  stations. 

The  central  angle  is  4  X  6  =  24°;  then,  from  Table  IV, 
c  =  595.74. 

By  (37), 

Arc  -  chord  =  1432.7  X  ^  -  595.74  =  4.34  ft. 

By  (39). 

.             ,      -       6X6X6X4X4  .._. 

Arc  -  chord  =  ^;r =  4.32  ft 

Remark.— Formula  (38)  is  interesting  as  showing  what  a  com- 


LOCATION. 


55 


paratively  small  iucrease  iu  length  of  liuo  is  caused  by  a  consid- 
erable lateral  deflection  in  alignment.  For  instance,  a  lateral 
deflection  of  2000  feet  is  made  at  the  mid-point  of  a  line  40,000 
feet  long  ;  what  will  be  the  increase  in  length? 

By  (38)  the   increase   is    )^^  ^^^J^    =  200   feet,    giving  for   the 


40,000 


increased  length  40,200  feet. 


B.  Locating  Simple  Curves. 

94.  To  Locate   a   Curve  with  the  Chain  by  Oft*ets   from 
Chords  Produced. 
In  Fig.  21  let  the  P.  C.  fall  at  B.    If  BC  is  a  full  chain,  prolong 


Fig.  21. 

the  tangent  AB  to  if,  making  55"=  BC ,  HO  will  equal  t,  which 
may  be  calculated  by  (32)  or  (33').  With  5  as  center,  strike  an 
arc  with  radius  BH,  and  with  H  as  center  and  t  as  radius  strike 
an  arc ,  at  G,  where  these  arcs  intersect,  set  a  stake.  Produce 
BC  to  K,  making  CK  =  BC  =  CD  \  strike  the  arc  ED  from  C  as 
center ;  make  the  chord  KD  =  d,  calculated  from  (29'),  (30'),  or 
(31).  Set  a  stake  at  D  and  proceed  in  like  manner  for  the  other 
points  until  the  P.T.  is  reached,  where  FP  is  made  equal  to  t. 

Usually  the  P.C.  does  not  fall  at  a  full  station  ;  then  HC  =  t', 
which  may  be  found  by  (34)  or  (35).  Using  this  value  of  t\  we 
locate  0  as  above.  At  B  make  BB  =  t',  and  prolong  RG  to 
L  ;  make  LB  =  t  and  set  a  stake  at  Z>.  EM  will  equal  d,  and 
may  be  located  as  before. 

We   may  regard   KD  as  equal  to  KL  -{-  t,  and,  finding,  KL, 


56        A   FIFLD-MANUAL   FOR   RAILROAD    ENGIXEERS. 

measure  KD  and  set  D  without  locatiug  R.     To  do  this  we  have 
the  similar  triangles  BHC  and  CKL,  from  which 

KL        t' 


CK     BC 
and  therefore,  since  KG  =  CD, 

In  like  manner  at  ^  we  have 

PN=  t^,    and    FP  =  W 
hence 

Make  EQ^  =  ti ,  prolong  QF,  and  wc  have  the  tangent  at  F. 
Example. — Given  the  P.C.  of  a  5"  curve  at  106  +  20  and  the 
angle  of  intersection  22°,  to  locate  the  curve. 

22 
Here  L  =  -^  =  4.4  stations. 

5 

Therefore  the  number  of  the  P.  T.  is 

106.20  +  4.4  =  sta.  110  -f  60. 

BCin  this  case  is  80  ft.,  and  by  (33') 

t  =  0.873  X  5  =  4.37  ft. 

By  (35),         ^=4.37X  (1^^=2.80  ft. 

Set  off  HC  -  2.80  ft.,  and  at  B  make 

KB  =  2.80  X  ^  +  4.37  =  7.87  ft. 

At  E  make  ME  =  d  =  8.72  by  (31).  This  will  be  at  sta.  109  ; 
at  110  set  a  stake  by  offsetting  8.72  ft.  The  last  chord  is  60  long, 
and  hence  the  offset 

NF=  4.37  X  Y^  +  4.37  X  '  ^^^)'=  2.62  +  1.57  =  4.19  ft. 
Make  .fi;^  =  1.57  ft  ,  and  prolong  QF,  the  terminal  tnngcnt. 


LOCATION". 


57 


H- 


95.  To  Locate  a  D  Degree  Curve  by  Oflfsets  from  Tangent. 

Let  AM,  Fig.  22,  be  tangent  at  A,  and  E,  F,  G,  etc.,  points  on 
tbe  curve.     Tbe  offsets  BE,    CF,     ^  B 

etc.,  may  be  found  from  formula  l 
(36), 

eitber  by  taking  equal  intervals,  ^ 
AB,  BC,  CM  along  tbe  tangent  or 
by  taking  E,  F,  O,  etc.,  at  regular 
stations  around  tbe  curve  and 
using  tbe  arc  lengtb  instead  of 
tbe  tangent. 

Wben  tbe  arc  AO  is  large,  or 
strict  accuracy  is  required,  we 
proceed  to  find  tbe  offsets  at 
regular  stations  and  tbe  lengtbs 
of  AB,  AG,  etc.  First  find  R 
from  (12)  or  (12');  tben  from  triangle  OEL, 


Fig.  23. 


BE=  AL  =  R{1  -  cos  D)  =  R  vers  D, 

AB  =  LE=  R  sin  D. 
lu  like  manner 

OF  =  AH  =  R{\  -  cos  22))  =  R  vers  2D, 
AC=  HF  =  R  sin  2D, 

and  so  on  for  any  number  of  stations. 

Sbould  A  fall  at  a  plus  station,  we  first  find  tbe  angle  Bi  at  tbe 
center,  tben 

BE  =  R  vers  Bi  , 

AB=  R  sin  P, , 
CF  =  R  vers  {Br  +  D), 
AC=  i?sin(A  +  B), 
etc.  =  etc. 

The  ordinates  BE,  CF,  etc.,  are  evidently  equal  to  tbe  mid- 
ordinates  for  long  chords  2LE,  2IIF,  etc.;  hence  we  can,  if 
A,  E,  F,  and  0,  fall  at  full  stations,  take  them  direct  from 
Table  V;  then  take  the  long  chords  from  Table  IV  and  dividing 
these  by  2,  get  the  required  coordinates. 


58        A    FIELD-MAJsUAL    FOR    RAILROAD    ENGINEERS. 


Example.— Locate  three  stations  of  a  4'  curve  by  offsets  every 
50  ft.  on  curve. 

Referring  to  Table  V,  the  required  offsets  are  0.87,  3.49,  7.85, 
13.94,  21.77,  and  31.31.  By  Table  lY  the  distances  measured 
along  tangent  are  50.0,  99.94,  149.76,  199.39,  248.78,  and  297.87. 
With  these  values  we  can  set  out  the  curve  either  way  from  A. 

Had  we  used  formula  (36)  we  should  have  had  for  the  values 
of  the  offsets  0.87,  3.50,  7.88,  14.00,  21.87,  and  31.50. 

96.  To  Locate  a  Curve  by  Offsets  from  a  given  Long 
Chord. 


Let  FK,   Fig.   23,   be  the  given  chord.     We  may  compute  the 
offsets  yi,y^. .  .Mhy  the  methods  of  83— of  which  formula  (17), 

y  =  ImnD, 

is  the  most  convenient,   within  the  limits  of  its  applicability— 
and  setting  off  these  ordinates,  locate  the  curve. 

Or  we  may  set  off  the  mid-ordinate  J/ =  i^^ersi^O^  at  A, 
and  at  C  set  off  ^2  =  iW  —  7?  vers  D,  making 

AC  =  HL  =  R  sin  D. 

GE  will  be 

yi  -  M  -  B  vers  2 A     find     AE  =  7?  sin  22). 

Anotheii  Method  is  to  find  the  angle  A'Oi^at  the  center,  and 
by   Table  IX  delennine  BA  -  M ;  then  by  Tables  V   and  IV 


LOCATION.  59 

delermino  BL,  BN,  Lll,  and  J^^G.     Then  HC  =  M  -  BL,  which 
scL  oil"  lit  C,  aud  other  points  in  like  manner. 

Example.— Given  the  P.C.  of  a  4°  curve  at  station  160  +  75, 
the  angle  between  tangent  and  chord  =  9°,  required  the  offsets 
necessai-y  to  locate  the  curve. 

Here  7=2x9  =  18°. 

18 
,'.     L  —  -r  =  4.50  stations. 
4 

Hence  the  P.T.  falls  at  160.75  +  4.50  r=  sta.  165  +  25.  The 
mid-point  on  curve  B  falls  at  sta.  163.     By  Table  IX, 


ri 


M='-^  =  17.64  ft. 
4 

By  Table  V  the  mid -ordinate  for  two  stations  of  a  4°  curve  is 

BL  =  3.49. 

Hence  HC  =  17.64  -  3.49  =  14.15. 

By  Table  IV,  IlL  =  AC  =  99.94  ft. 

Measure  AG  =  99.94  ft.,  and  set  off  CII=  14.15  ft.,  and  drive  a 
stake  at  //.     In  like  manner  lind 

GE=d.lO    and     ^1^=  199.39  ft. 

The  points  P  and  Q  are  also  located  by  means  of  the  coordi- 
hates  just  determined. 

If  B  had  fallen  at  an  odd  station,  the  curve  could  have  been 
located  in  the  same  manner,  7/ and  P  being  100  ft.  from  B,  G  and 
Q  200,  etc. 

97.  To  Locate  a  Curve  with  Transit  and  Chain  when  the 
Degree  D  or  Radius  E  is  Known. 

If  R  is  given,  determine  7>  by  (13);  then,  since  the  angle  in 
the  circumference  of  a  circle  is  half  the  angle  at  the  center  sub- 
tended by  the  same  chord,  we  may  locate  points  on  the  curve  by 
successive  deflections  from  the  tangent. 

In  Fig.  24  let  the  P  C.  be  at  A,  at  which  point  set  the  transit, 
and  with  the  vernier  plates  clamped  at  zero  place  the  telescope 
in  tangent  either  by  sighting  the  P.I.  or  by  backsighting  to  some 
point  in  the  tangent  Deflect  from  the  tangent  half  the  angle  at 
the  center  for  the  sub-chord  or  chord,  and  direct  the  head  chain 
man  into  line  while  the  real  chainnian  holds  his  end  of  the  chain 


60        A    FIELD-MANTAL    FOR    RAILROAD    ENCtINEERS. 

at  Ihe  transit,  the  cbaiu  beiug  kepi  taut.  The  stakeuiau  drives  a 
stake  at  the  point  where  the  head  chainman's  Q.i\g  rested,  aud  the 
rear  chainman  advances  to  this  point.  Deflect  iD  from  the  chord 
AB  just  run,  aud  while  the  rear  chainman  holds  his  end  of  the 
chain  at  B  direct  the  head  chainman  into  line  at  C.  Other  points 
are  located  by  deflecting  an  additional  ID  for  each  chord  length 
measured,  until  a  point  E  is  reached  to  which  it  is  desirable  to 


Fig.  '-M. 

move  the  transit.  The  angle  FAE  shouM  not  exceed  about  15°. 
Move  the  transit  to  E,  backsight  to  A,  and  deflect  FEA  =  EAF, 
when  the  telescope  will  be  in  tangent,  and  the  curve  can  be  con- 
tinued until  it  is  again  necessary  to  move  the  transit.  At  the 
P.T.  put  the  telescope  in  tangent  by  backsighting  to  the  point 
last  occupied  by  transit  and  deflecting  the  tangential  angle  as  at 
E.     The  line  may  now  be  continued. 

98.  The  Index-angle  is  read  on  the  vernier-plate,  and  is  the 
angle  between  the  tangent  to  the  curve  at  the  P.C.  and  any  other 
line  passing  through  a  point  on  the  curve  when  the  telescope  is 
directed  along  this  line.  It  is  most  frequently  taken  as  the  angle 
between  the  initial  and  any  subsequent  tangent  to  the  curve. 
Thus  at  E  the  index-angle  equals  EFP  =  2FAE.  At  any  point 
on  the  curve  the  index-reading  in  tangent  may  be  found  by  the 
following  rule,  which  may  be  easily  deduced  from  a  figure: 

From  double  ihe  index-angle  that  fixed  the  point  subtract  the  index- 
angle  in  tangent  at  the  last  point;  the  remainder  is  the  index-angle 
required. 

99.  Subdeflection-angles  may  be  found  by  (13)  ligidly,  or 
approximately  (and  with  sufficient  accuracy  except  when  D  is  very 
large)  by  assuming  the  central  angles  to  be  proportional  to  their 
chords.  Thus  on  a  4°  curve  the  central  angle  for  a  sub-chord  of 
25  ft.  would  be  V,  and  the  subdefleclioiianirle  30'. 


LOCATIOK.  61 

Example. —Locaie  a  V  curve  to  left  when  the  P.C.  is  at  sla. 
81  -f- 25  and  I=H2'  86'. 

32.6 
Here  L  -  — ,"—  =  8.15  chains. 

4 


Hence  the  P.  T.  will  fa.l  ac  81.25  -j-  8.15  =  sta.  89  +  40.     The 
rst  sub-chord 
found  by  (13). 


first  sub-chord  is  75  ft.  lon^,  and  the  tirst  deflection-angle  will  be 


^'°^*  =  im7-»»2«" 


i(5  =  V  30'. 


By  the  approximate  rule,  since  ^D  =  2", 

45  _  75 
2        100' 

whence  i5  =  2  X  I  =  T  30'  as  before. 

With  transit  at  P.C.  deflect  1°  30'  from  tangent,  measure  75 
feet,  and  set  stu.  82.  Then  a  deflection  of  3'  30'  will  determine 
83,  5"  30'  sta.  84,  7°  30  sta.  85.  Now  remove  transit  to  85,  and 
with  vernier  at  7°  30'  backsight  to  81  +  25.  Reverse  telescope 
and  set  vernier  at  15°  00',  when  the  telescope  will  be  in  tangent. 
An  index  angle  of  17'  will  fix  86,  and  so  on. 

The  last  chord  will  be  only  40  feet  long,  for  which  the  sub- 
deflection-angle  is  {§^  of  2^  that  is,  48'.  The  index-angle  fixing 
the  P.T.  is  therefore  23°  48'. 

To  get  in  tangent  at  89  -f  40  backsight  to  sta.  85,  with  vernier 
at  23°  48' ;  then  by  the  rule  of  98  the  index-reading  is  (23°  48')  X 
2  —  15'  =  32'  36'  =  /.  Set  the  vernier  at  this  reading  and  run 
tangent. 

Caution  —It  is  not  good  practice  to  set  more  than  4  or  5  sta- 
tions on  curve  from  any  one  point.  Mr.  Siiunk  gives  the  limit- 
ing angle  to  be  deflected  from  tangent  as  20%  and  sa^s  15°  should 
rarel}'  be  exceeded.     {Field  Engineer,  p.  82.) 

100.  The  Transit  Notes  may  be  conveniently  kept  in  the  form 
below,  which  shows  the  notes  for  the  last  example. 

When  possible  the  tangents  should  be  run  to  intersection,  the 
angle  1  measured,  and  the  t:ingent  distance  calculated.     Then 


62        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS 


a 

O     - 

,  ii) 

■a 

'Z  a> 

•z  ® 

X  c 

y.  ^ 

cs  j2 

Oj   03 

Station. 

C   TO 

i-iH 

II 

Remarks. 

90 

+40 

QP.T. 

0°48' 

23°  48' 

32°  36' 

N  37°36' E 

N  27°30'  E 

89 

23°  0' 

88 

21°   0' 

87 

19°  0' 

86 

17°  0' 

85 

O 

7°  30' 

15°  C 

84 

.5°  30' 

83 

2°    0' 

3°  30' 

82 

1°30' 

1°30' 

4°  C.L.;  P.I.  set. 

-f25 

OP.C.4°C.L. 

0°    0' 

0°  0' 

0°  0' 

I  =  32°  36';     T  ~ 
418.9  ft. 

81 

N  60°12'  E 

N60°10'E 

measure  along  tangents  and  set  P.C.  and  P.T.  from  the  P. I. 
When  the  curve  is  run  in,  the  position  of  the  P.T.  thus  found 
should  agree  with  the  one  set  from  the  P.I.  If  the  error  is 
greater  than  the  circumstances  of  the  case  permit,  the  curve 
must  be  rerun  and  tanircnts  remeasured. 


101.  Another  Form  of  Notes,  and  in  some  respects  a  better  one 
than  the  above,  is  given  below.  Tl^e  index-readings  are  com- 
puted as  though  the  entire  curve  v^-ere  run  from  the  P.C.  The 
notes  for  ihe  last  example  would  appear  as  below  : 


Station. 

otal 
ngle. 

3  5 

Remarks. 

M  p 

E-<J 

■5^ 

cS  "^ 

Q 

O 

90 

+40 

0P.5. 

0°48' 

]G°1S' 

32°  36' 

N  27°36'  E 

N27°30'E 

89 

15° 30' 

88 

13°  30' 

87 

11°30' 

86 

9°  30' 

85 

O 

7°  30' 

84 

5°3C' 

83 

2°   0',  3°  30' 

82 

1°30'    1°30' 

4°  curve  left; 

+25 

OP.a4°c.L. 

0°    0' 

0°  0' 

P./.  set.  /=32°36/; 
3'=  418.9  ft. 

81 

N  G0°12'  E 

N  00°  10'  E 

The  computations  are  all  made  before  beginning  the  work,  and 
the  notes  have  the  advantage  of  permitting  the  tracing  of  the 
curve  either  way  from  the  instrument  without  additional  compu- 


LOCATION.  63 

tatioDS.  Suppose  the  tmnsilinau  to  have  niu  the  curve  from  the 
P.C.  to  sta.  85,  to  which  point  he  removes  the  instrument.  He 
there  sets  the  vernier  at  0" — the  angle  on  limb  when  telescope 
was  in  tangent  at  the  P.C. — then  sighting  theP. C.  he  reverses 
the  telescope  and  deflects  to  9"  30  ,  which  will  fix  sta.  86.  Had 
the  tangent  at  85  been  desired,  a  reading  of  T  30' — the  angle  that 
located  that  point — would  have  put  the  telescope  in  the  plane  de- 
sired. A  reading  of  11°  30  fixes  87,  and  so  on  to  the  P.T. 
Removing  to  the  P.T.,  the  plates  are  clamped  at  T  30',  and  a 
backsight  to  sta.  85  taken  ;  then  deflecting  to  16°  18',  the  tele- 
scope is  in  tangent  at  the  P.  T.  Had  it  been  desirable  to  set  84 
from  85,  a  reading  of  5°  30'  would  fix  that  point ;  others  may 
be  found  in  the  same  manner. 

•Any  convenient  form  of  notes,  which  are  intelligible  to  another 
engineer  who  may  have  to  retrace  the  curve,  may  be  used,  but  it 
is  desirable  that  some  general  form  should  be  emploj^ed.  Either 
of  the  preceding  forms  seems  to  meet  ordinary  requirements. 

C.  Obstacles. 

102.  To  Pass  an  Obstacle  on  a  Curve. 

First.  Suppose  the  obstacle  to  he  one  obstructing  vision  at  one 
station  only. 

In  Fig.  25  suppose  transit  set  at  A,  and  B  and  C  located  from 
that  point,  but  the  next  full  station,  H,  to  be  invisible  from  A. 


Fig.  25. 

Set  a  plus  station  at  E,  as  near  the  obstruction  as  may  be  conven 
lent,  then  set  i^lOO  feet  from  E.     Next  make  FG  -  100  -  GE, 
and   locate  G  with  the   corresponding   deflection-angle.     Other 
stakes  may  be  set  beyond  G,  or  the  transit  may  be  removed  to 
that  point  and  the  curve  beyond  traced. 

Second.     Suppose  the  line  of  sight  obscured  for  more  than  one 
station,  as  in  Fig.  20. 


64 


A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


If  trausit  is  nt  A,  deflect  an  angle  HAB  that  will  clear  all  ob 
structions,  and  at  the  same  time  cause  B  to  fall  at  a  full  statiou. 
Then  by  Table  IV,  Table  IX,  or  by  formula  (16)  calculate  the 
long  chord  AB  ;  measure  AB  and  move  transit  ioB  ;  then  deflect 


Fig.  26. 

the  angle  ABC  =  BAR  when  the  telescope  will  be  in  tangent. 
The  curve  may  now  be  run  both  ways  from  B. 

If  it  happen  that  some  stations,  as  E  and  F  in  the  figure,  arc 
still  invisible,  they  may  be  located  by  offsets  from  chord  or  tan 
gent. 

Example. — Let  the  curve  be  a  3°  curve  to  right ;  angle  HAB 
=  T  30',  the  deflection-angle  for  5  stations.  By  Table  IV  the 
long  chord  is  498.63  feet,  which  can  now  be  measured  and  a  hub 
set  at  B\  then  making  angle  CBA  =  7°  30',  the  telescope  will  be 
in  tangent  and  the  curve  can  be  traced  either  way. 

103.  To  Locate  a  Curve  when  the  P.  C.  is  Inaccessible. 

^  In  Fig.  27  let  the  P.  C.  at  B  be  in- 

accessible ;  it  is  desired  to  reach  ^ 
-  point  H  on  accessible  ground. 

First  Method,  —  Assume  a 
point  iZ^  on  the  curve  such  that  a 
line  AH  from  an  accessible  point 
A,  on  tangent,  will  clear  the  ob- 
stacle ;  for  convenience  H  should 
be  at  a  full  station.  The  arc  BH 
and  central  angle,  which  equals 
HCF,  are  then  known.  Calculate 
BC  =  T  hy  (14)  or  (14«)  ;  then 
since  AB  is  known,  AC,  =  AB -t 
BC,  is  known. 

Now  in  triangle  J.  CiET,  from  trig- 
onometry, 

tan  |(7^  -  a)  _  AC  -  CH 
tan  lih  +  a)  ~  AC-\-  CH" 


Fig.  27. 


LOCATION.  65 

But  (h  -\-  a)  =  c;  hence 

tau  liJi  -  a)  =  ^^  "    ^,^  tan  ^c (40) 

Then  l{h  +  a)  +  l{7i  —  «)  =  h,  tbe  larger  angle,  and 
|(7i  +  a)  —  l{7i  —  a)  =  a,  the  smaller  angle.  AH  may  be 
found  by  the  law  of  sines,  or  by  drawing  CE  perpendicular 
to  All,  when 

^^=  .4C'cos«  4-  Ci/cos7i (41) 

Example.— The  P.C.  of  a  4°  curve  is  at  sta.  141  +  25,  aud  it 
is  desired  to  reach  the  point  i/froni  sta.  139  on  tangent. 

Suppose  ^be  assumed  to  fall  at  sta.  147  ;  the  curve  length  is 
i  =  147  —  141.25  =  5.75  cbaius.  Then  angle  c  =  5.75  X  4  = 
23"  0'.  By  Table  IX  the  tangent  distance  for  a  T  curve  is 
Tx  i-  23"  =  1165.8  ft. 

,„  1165.8  ^ni      tr-   i?i 

By  (14a),  T  =  —^  =  291.45  ft. 

Now  AC  =  291.45  +  225  =  516.45  ft., 

and 

AG-{-  GH=  516.45  +  291.45  =  807.90, 

while 

AC  -  CE=  225  ft. ; 

hence,  by  (40), 

tan  Uh  -a)  =  -^  -  X  0.20345  =  0.05666  =  tan  3"  15'. 

80  <  .y 


Therefore 


and 


7i  =  ir  30'  +  3°  15'  =  14°  45', 
a  =  11°  30'  -  3°  15'  =    8°  15'. 


By  (41), 

AH=  516.45  X  0.98965  +  291.45  X  0.96705  =  793.0  ft. 

At  A  deflect  8°  15'  from  tangent,  measure  793.0  ft.  and  seta 
hub  ;  move  to  this  point,  backsight  to  A  and  deflect  14°  45'  into 
tangent,  then  trace  in  the  curve. 


66 


A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


Second  Method. — If  F,  auy  assumed  point  in  tangent,  is 
visible  from  A,  AF  may  be  measured  by  some  indirect  method; 
then  AF  —  AB  =  T.  The  tangent  for  a  1°  curve  having  same 
intersection-angle,  KFG,  is  Ti  =  T  X  IJ' ;  find  this  value  of  Ti  in 
Table  IX  and  take  out  the  corresponding  value  of  I.  With 
transit  at  F  deflect  the  angle  KFG,  measure  FG  =  FB  =  T,  and 
set  hub  at  G.  The  station  number  of  G  is  found  by  dividing  the 
central  angle,  =  KFG,  by  the  degree  of  curve  D.  Move  to  G  and 
trace  the  curve. 

Example. — Let  AF  measure  490.5  ft.  from  sta.  139  of  the  last 
example.  Then  AB  =  235  ft.,  and  BF=  490.5  -  225  -^  265.5  ft. 
265.5  X  4=  1062  ft.,  which  by  Table  IX  is  the  value  of  Ti  for 
1=  2V.    Set  transit  at  F,  deflect  2V,  and  measure  FG  =  265.5  ft. 


L  =  -j-=  o.2d  chams; 
4 


hence  (?  will  fall  at  141.25  +  5.25  =  sta.  146  +  50.     Move  to  O 
and  run  the  curve  both  ways. 

Third  Method. — In  Fig.  28  let  the  inaccessible  P.C.  be  at  B, 
and  let  it  be  required  to  reach   E  from  a  point  0  on   the  curve 

prolonged  backwards  from  B. 

At  a  given  point  A  on  tangent  cal- 
culate the  tangent  offset  by  (36)  or 
the  methods  of  95,  then  set  this  off  at 
right  angles  to  AB  ;  set  the  transit  at 
C  and  turn  oS  ACL  =  90°  -  COB, 
when  the  telescope  will  be  in  tangent 
at  C.  COB  may  be  found  from  Table 
IX  by  multiplying  AC  by  the  degree 
of  curve  and  taking  half  the  intersec- 
tion-angle corresponding  to  the  mid- 
ordiuate  that  equals  this  product.  Now  deflect  and  measure 
ECL,  then  by  (16)  or  (16(0  calculate  CE,  which  measure.  Move 
to  E  and  deflect  LEC  =  ECL  and  the  telescope  will  be  in 
tangent.  The  central  angle  BOE  =  2LEC  -  BOC,  from  which 
the  arc  BE  and  number  of  sta.  E  may  be  found. 

Example. — Take  the  same  example  as  in  the  last  two  cases. 
.4  is  at  sta.  139,  B  at  141  +  25;  hence  AB  =  2.25  stations. 


Fig.  28. 


By  (36),        z  =  AC=iX  (2.25)'^  X  4  =  17.72  ft. 


LOCATION. 


Or  by  Table  IX  the  angle  corresponding  to  the  long  chord 
(2  X  2.25)  X  4  —  1800  ft.  is  18'  4',  for  which  the  mid-ordinate  is 

71.06  ft.     For  our  4'  curve  the  mid-ordinate  will  be  — '- —  =  17.77 

4 

ft.,  which  equals  AG  and  agrees  closely  enough  with  the  value 

for  z  above. 

Make   angle  5.46' =90'.  and   measure  .16' =  17.72  ft.     Move 

to  C  and  sight  to  A.  then  make  angle  .46'Z  =  90"  -  (9°  2^)  = 

80'  58'.     Suppose  an  angle  LCE  —  16'  1'  to  clear  the  obstacle. 

By  formula  (16), 

CE=2R  sin  (16°  1')  =f  2  X  1432.7  X  0.27592  =  790.6  ft. 

Measure  along  CE  790.6  ft.  and  set  a  hub;  move  to  E  and  run 
the  curve. 

CE  might  have  been  found  by  means  of  Table  IX,  for  the  long 
chord  of  a  V  curve  having  i  =  2LCE=  32' 2'  is  3162.0  ft.; 
divide  this  by  4  and  there  results  CE  =  790.5  ft. 

104.  To  Pass  to  Tangent  when  the  P.2\  is  Inaccessible. 

This  is  just  the  reverse  of  the  preceding  problem,  and  may  be 
•iccomplished  by  reversing  the  processes  described  above. 

When  the  P.T.,  however,  falls  iu  or  beyond  a  river  or  lake 
obstructing  the  ordinary  methods  of  indirect  measurement,  the 
case  merits  a  special  solution. 

First  Method  —In  Fig.  29  let  the  transit  be  at  A,  and  B  the 
P.T.      From    the    known   station 
numbers  of  A  and  B  the  length  of 
curve  and  angle  /  may  be  found; 
then,  by  (14),  AC  =  R  tan  \I,  or, 

hy  {Ua),  AC  =  ^^^^. 

Move  to  C  and  deflect  the  angle 
7;  set  a  stake  F,  and  one  at  some 
other  accessible  point  E,  measure 
anorle  ECF=c.  Move  to  -F  and 
measure  the  angle  EFC  and  the 
side  EF:  then  in  triangle  ECF 
angle  e  =  \80-  {c  -\-f);  by  trigo- 
nometry 

CF=^^.EF. (42) 

sin  c 


Fig.  29. 


68        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

Since  BC  =  AC,  there  results  BF=  CF  —  AC;  and  as  the  sta- 
tion number  at  B  is  known,  that  at  F  becomes  known,  and  the  line 
may  be  continued. 

If  B  is  not  the  P.T.,  measure  back  the  distance  FB,  set  transit 
at  B,  and  continue  the  curve. 

Example.— Let  the  P.T.  of  a  2°  C.L.  fall  at  sta.  205  +  5C— an 
inaccessible  point;  suppose  A  at  sta.  200,  angle  c  =  40°,/=  80% 
EF  =  310  ft. 


Here 


7=5.50  X  2=  ir  0',     and      «  =  60° 


By  (14a),         T  =  — ^—  =  275.87  ft. 


From  (42),  applying  logarithms. 


log  CF=  2.49136  -f  9.93753  -    9.80807  =  2.6082. 


Whence  C'i^=  417.7  ft.     Then   ^i^*-- 417.7  -  275.87  ==  141.8  ft.; 

therefore  the  number  of  i^will  be  206  +  91.8. 

Second  Method.— In  Fig.  30,  with  the  transit  at  any  point  A 

on  the  curve,  assume  a  long  chord  AB 
and  calcuhite  the  angle  CAB;  deflect 
this  angle  from  the  tangent  AC,  and  set 
a  point  E  beyond  obstruction ;  set  also 
a  stake  at  C  in  tangent. 

Move  to  E  and  measure  A  EC  and 
side  EC.  Compute  AE  from  the  trian- 
gle A  EC.  If  this  is  greater  or  less 
than  the  length  of  the  long  chord  AB, 
take  their  diJSerence  BE  and  set  a  hulj 
at  B.  With  the  transit  at  B  trace  out 
the  curve. 

Example. — Given  A  at  sta.  210  of  a 
3"  C.  L.,  angle  a  =  12°.  b=  92°,  EC 
=  181  ft.     Then  c  =  76°,  and  by  solving 

the  triangle  A  EC,  AE=  844.7  ft.    By  Table  IX  the  long  chord  of 

90Q9  g 

a  r  curve  for  /=  24°  is  2382.6  ft. ;  therefore  AB  =  '^—^  =  794.2 

ft.     Now  will  ^Z?  =  844.7- 794.2  =  60.5  ft.,  which  is  the  dis- 
tance iilong  EA  that  transit  must  be  moved  back  from  E. 


Fig 


LOCATION. 


69 


105.  Given  the  Perpendicular  j)  from  a  Point  to  a  Tangent, 
to  Find  the  Point  on  Tangent  at  which  to  Begin  a  Curve  of 
Given  Radius  which  will  Pass  through  the  Given  Point. 

First  SohUTiON.— In  Fig.  81  let  F  be  the  point,  BPi\iQ  per- 
pendicular.     We    have    to    find 
BA  =  X. 

From  P  draw   PC  parallel   to 
AB ;  then  in  triangle  OPG 

JB2  =  iT-^  +  (i2  -  pf. 
From  which 


X  =  i/21ip  -  p\      .     (43) 

Second  Solution.— Consider 
p  =  AC  as  the  mid -ordinate  for 
a  long  chord  =  2x  :  then  pX  B 
=  the  mid-ordiuate  for  a  1°  curve 
for  a  central  angle  equal  2a. 
The  corresponding  long  chord  may  be  taken  from  Table  IX. 
Then 


Fio.  31. 


_  IL.Ci 


(43a) 


Example.— Given  p  =  30  ft.,  Z)  =  4*  (i?  =  1432.7),  to  find  x. 


By  (43),         X  =  V85,962  -  900  =  291.65  feet. 
By  the  second  method, 

30  X  4  =  120, 


the  mid-ordinate  for  a  1°  curve  corresponding  to   an  angle   of 
23°  29',  for  which  the  long  chord  is  2332.6.     Now.  by  (43a), 


1  ^^  2332.6       ,^^.  „  .    ^ 
a;  =  -  X  — i —  =  291.6  feet. 

2  4 


106.  In  Fig.   31,  Given  x  and  p  to  Find  the  Radius    of  a 
Curve  Tangent  to  A  B  sit  A  and  Passing  through  P. 


From  r43), 


R  = 


x'^  +  P'' 


2p 


(44) 


70        A   FIELD-MANCAL   FOR   RAILROAD   ENGIXEERS. 

107.  Given  the  Location  of  a  Point  P  referred  to  the  P.  I. 
to  Find  the  Radius  of  a  Curve  through  P  which  will  Unite 
the  Given  Tangents. 


Fio.  32. 


In  Fig.  32  suppose  BC  =  I,  BP  =  m  known,  and  angle  a  cal- 
culated ;  or  PC  and  a  may  be  measured  ou  the  field. 
From  triangle  CAO, 

J  =  90'  -  (a  +  \I),     and     CO  =  i2  sec  \I. 


Now  from  triangle  PCO, 


CO    .    . 
sin  y  =  —  sm  b. 


Inserting  values  of  PO  and  CO, 


B  sec  ^I  1  7     •    r        sm  & 

sm  y  =  ---■—  .  s\u  b  =  sec  hi .  sm  o  = —=r, 

R  '  cos  hi 


(45) 


an  equation  from  which  the  imknowu  R  has  disappeared.     Next, 
from  the  same  triangle,  since  x  =  180°  —  {h  -\-  y), 


ij  =  !^_* .  PC 


sm  X 


(46) 


When  I  =  90°,  it  can  easily  be  shown  that 


n  =  I  -^  m  -\-  i2lm (47) 


LOCATION'. 


Tl 


108.   To    Locate    a   Tangent    to    a    Curve    from   an  Outside 
Point. 
FiBST  Method. — In  Fig.  33  let  P  be  the  point  and  AHB  iLe 


Fie.  33. 

curve.  Run  a  trial-line  PA  cutting  the  curve  in  A  and  B. 
Measure  PA  and  AB :  or  measure  PA  and  angle  a  between  the 
chord  AB  and  tangent  AL.     Then 

AB  =  2AC  =  '2Bsma, 
OC  =  R  cos  a. 


By  geometry,  PE  =  *  PA  x  PB,  PE  being  the  required  tan- 

CO 


gent.     From  the  figure 


tan  n  = 


tan  m  = 


CP 
PE' 


At  P  deflect  the  angle  i  =  m—n  from  PA  and  run  the  tangent. 
Second  Method. — In   Table  IX  find   the  long  chord  for  t 
central  angle  2<i  ;  then 


AB  =  2AC  = 


~~B~' 


CH  = 


M, 


D  * 

and  CO  =  B-  CH. 

We  may  now  proceed  as  before. 


"/O 


i4        A    FIELD-MAXUAL   FOR    RAILliOAD    ENGINEERS. 


109.  To  Run  a  Tangent  to  Two  Located  Curves  of  Contrary 
Flexure. 

First  Case.— Iu  Fig.  34  let  FK  and  LB  be  the  curves,  and 
-SX  =  p  measured  on  the  ground. 


Fia.  34. 

Let  FE  =  t  be  the  required  tangent. 

Draw  OiH  parallel  and  OM  perpendicular  to  FF ;  from  the 
triangle  O.HO^ ,  since  FIT  =  R, , 


whence 


t=   |/2(i?i  4-i?,)^-f_p2. 


Also, 


cos  a  = 


Ri  -\-  Ri 


Ri-\-R'i-\-p 


(48) 


(49) 


The  arcs  FK  and  LE  may  be  found  from  the  angle  a  and  the 
known  curvatures,  after  which  the  points  i^  and  -fi'may  be  set. 

If  t  is  given  and  p  required,  it  may  easily  be  found  from  (48). 

Second  Case,    p  not  known. 

Set  the  transit  at  a  point  A  on  one  curve  and  note  the  bearing 
of  the  tangent  to  the  curve  at  that  point  (see  Fig.  34);  the  bearing 
of  the  radius  O-iA  differs  from  this  by  90°.  Run  a  line  ABC  of 
one  or  more  courses  to  intersect  the  other  curve  at  C.  Kote  the 
bearings  and  lengths  of  these  courses  and  the  bearing  in  tangent 
at  G,  from  which  calculate  the  bearing  of  W,.  R,  and  R.^  being 
known,  the  latitudes  and  departures  are  next  calculated.    Let  O-jiV 


LOCATION. 


73 


be  the  sum  of  the  northings  or  southings,  0,iVthe  sum  of  the 
eastings  or  westings  ;  from  tlie  triiingle  OxO-iN, 


tan  h  = 


0,N 


and 


OxOa  =  VOxN"-{-0,N\ 


As  before,  FE  is  the  required  tangent  and  O2  H"  perpendicular, 
while  Oii/ is  parallel  thereto. 


cos  a  = 


Ri  +  R^ 
0,0, 


Angle  FO,N=  J  -  a  is  the  bearing  of  O^F,  while  AOaF  = 
c  ^  b-\-  a  is  the  angle  of  retreat  f lora  the  known  point  A  to  F, 
where  the  tangent  may  be  run.     The  length  of  t  =  OiHis 

t  =  OOi  sin  a. 


D.   Change  of  Location. 

110.  To  Locate  a  Curve  Parallel  to  a  Given  Curve. 

Let  p  be  the  perpendicular  between  parallel  tangents,  and  sup- 
pose ABC  located  (see  Fig.  35). 
If  there  are  no  restrictions  as  to  the 
position  of  the  poiLts  E,  F,  and  O 
on  the  second  curve,  we  may  cal- 
culate the  new  degree  of  curve  Di 
for  a  radius  Ri  =  R  -}-  p,  by  (13), 
and  trace  the  curve  from  any 
poiut,  as  E.     Thus 

If,  however,  points  on  the  radii 
through  A,  B,  and  C  are  wanted, 
they  are  gotten  by  using  the  same  degree  of  curve  D  and  com- 
puiiug  the  length  of  chord  FE.     From  similar  triangles, 


EF 
Ri 


AB 
R 


100 
R' 


74        A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


whence 


EF  =100^  =  100  5^  =  100(l  +  I).  .     .     (50) 


Had  EFO  been  the  located  curve,  with  radius  B,  we  should 
have  had 


AB  =  100 


B  —  p 


(51) 


111.  To  Change  the  P.C.  of  a  Located  Curve  so  that  P.T. 
will  Fall  in  a  Given  Tangent  Parallel  to  Terminal  Tangent  of 

Located  Curve. 

Let  AB,  Fig.  36,  be  the  lo- 
cated curve  ;  FE,  the  tangent 
iu  which  the  P.T.  must  fall. 

Let  the  distance  between  tan- 
gents be  HE  =  p. 

Draw  BE  and  00'  parallel  to 
^2^;  evidently  ^(7  =  00' =^£', 
0'  being  the  new  position  of 
center. 


In  triangle  BEE, 


BE=  AC  = 


P 


sin  / 


=  p  cosec  /. 


(52) 


Set  the  new  P.C.  by  measurement  from  A,  and  run  the  curve 
CE.  Any  system  of  straight  lines  and  curves  may  be  treated  as 
above,  provided  I  is  the  angle  between  initial  and  terminal 
tangents  and  p  as  before. 

Example. — A  located  2°  30'  curve,  having  I  =  25°,  ends  in  a 
tangent  25  ft.  outside  of  desired  tangent.  Find  the  change  in 
position  of  P.  C. 


By  (52), 


^0  =  25  X  2.36620  =  59.16  ft. 


LOCATION. 


75 


112.  To  Find  the  Change  in  Radius  and  Position  of  P.C.  if 
P.T.  is  Required  to  fall  on  the  same  Radial  Line  but  on  a 
Tangent  distant  p  from,  and  parallel  to,  Terminal  Tangent  to 
Located  Curve, 

lu  Fig.  37  let  AB  be  the  located  and  CE  the  required  curve. 
Draw  the  parallel  chords  jiB  and 
GE.  Draw  Ci7 and  i?7*^ perpendicular 
to  AB.  The  angles  FBE^  CAH^U 

From  the  figure, 

CH=  AC  sin  il 
BF  —  BE  cos  hi  =  p  cos  \L 
Equating, 

^  C  sin  hi  =  p  cos  ^/, 
whence 


A                C 

K 

L 

^vT — ^-~^' 

X 

"V^ 

\          ^ 

\ 
X 

^ 

P 

< 

^ 

'  V"^ 

0 

Fig.  37. 


AG  =  p  cot  \L 


(53) 


In  the  triangle  OPO,,  0,P  =  AG,  OP  =  R  -  B^,  and 
{B  —  Bi)  ian  I  =  AC  =  p  cot  ^/, 
or  B  —  Bi  =  AC  cot  I  =  p  cot  ^I .  cot  /. 

Therefore 

Bi  =  R  -  AGcotI=  B-  pcot^I.  cot  I.  . 
From  trigonometry, 


cot  i7= 


-S-' 


sin  I 
1  —  cos  / 


and    cot  /  = 


cos  / 

sin  r 


(54) 


Inserting  these  values  in  (54)  gives 

n        n              sin  7        cos  /      „            cos  /  „         cos  I 

Bi  =B  ~p  . ^3^;^, .  -jz-F  —  B  —  p , Y-  B—p- 


1  -  cos  7 '  sin  / 
From  trigonometry,  ex  sec  7  = 

.-.  Bi  =  B  - 


1  —  cos  7 


vers  7 


vers  7 


cos  7 
P 


ex  sec  7 


(54) 


76 


A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


Example. — A  2'  30'  curve  strikes  25  ft.  inside  a  tangent  in 
which  the  P.T.  must  fall.  Find  the  necessary  change  in  radius 
and  position  of  P.C.  when  /  =  25''. 

By  (53)  the  change  in  P.  G.  is 

^a  =  25  X  4.51071  =  112.77  ft. 


By  (54'), 


Bi  =  2292.01  - 


25 


.10338 


=  2050.38  ft. 


By  Table  I  we  find  this  to  be  the  radius  of  a  2°  47'  41"  curve. 

113.  Given  a  Located  Curve  uniting  Two  Tangents  to 
Find  the  Change  in  Position  of  P.  C.  or  in  Radius  for  a  Given 
Change  in  the  Intersection-angle. 

First  Case, — Radius  unclianged. 

E^  In  Fig.  38  let  BCE  =  /  be  the  origi- 

.  .,  nal  intersection-angle,  FCE  =  /'  the 

'!  I  new  angle.     From  the  figure, 

AG  =  AG  -  GG, 


or 


AG  =  R  (tan  |7  -  tan  U').    (55) 


By  Table  IX. — From  the  table,  foi 
angle  /, 


r  = 


D 


Fia.  38. 


For/' 


T'  = 


Then 


AG=T  -  r. 


Second  Case.— P.C  unchanged. 

Here  the  tangent   T  for  the   two  curves  is  the  same,   and 

therefore 

R,  tani/'  =  i?tanU; 


Whence 


R,  =  i?tan  i/.cot  U'. 


(56^ 


LOCATION. 

By  Table  IX, 


D  Di 

whence  Di  =  ; .  U  = rr — • 

114.  To  Find  the  Change  in  R  and  P.  C.  for  a  Given  Change 
in  /,  the  P.T.  remaining  unchanged 


'0, 
Fig.  39. 

In  Fig.  39,  from  the  triangles  OBG  and  OiBH, 

OG  =  E  cos  / 

and  OiH  =  Ri  cos  /i. 

Now  OA  =  EF\  hence 

7?.  —  Ri  cos  Ii  =  R  ~  R  cos  /. 

Whence 

a  =  ijl^l^"^  =  r'-^ (57) 

1  —  cos  7i  vers  /i 

Also,  FA  =  HQ  =  BE  -  BG. 

Inserting  values  of  BE  and  BG,  there  results 

FA  =  Ri  sin  ii  -  i?  sin  /. (58) 

115.  Given  a  Located  Curve  to  Find  the  Change  in  R  for 
a  Given  Change  in  T,  I  remaining  unchanged. 

In  Fig.  40,  from  the  triangles  OAC  and  OiEG,  since 
EA  =  EC  -  AG, 


78        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


Whence 


lii  tan  il  -  H  tan  il  =  EA  =  T' 

c 


(59) 


Fig.  40. 


By  Table  IK.—EA  being  known,  T'  =  T-\-  EA.    Then,  by 
(15a), 


D,  = 


T' 


If  the  change  in  vertex  of  curve  is  wanted,  there  results,  from 
(25). 

E=  CG  =  Tian  K       E'  =  CH  =  T'  tan  J/. 
Therefore         0H=  E'  -  E={T'  -  T)  tan  il.     .     .     .     (60) 

GH  can  be  found  from  Table  IX  after  finding  Z),  as  above. 
If  Bi  is  given  and  EA  wanted,  (59)  yields 

EA=  T'  -  T=z  (R,  -  R)  tan  \L 

116.  To  Find  the  Radius  of  a  Curve  having  the  Same  P.O. 

as  a  Given  Curve,  but  ending  in 
a  Parallel  Tangent. 

In  Fig.  41  let  the  perpendicular 
distance  between  tangents  be/),  and 
AB  be  the  located  curve;  AOi  =  Ri 
is  required. 

FiKST  Method.  —  Draw  OH  aC 
right  angles  to  OiE;  then 

OiE=  0,H-\-  HG  -^  GE, 
or 

R,  =  (Ri  -  R)  cos  I -}-  R -\-  p. 

1 ^ r=^+-^f'    '     •     •     (61) 

1  —  cos  I  vers  /  ^ 


Fig.  4L 
From  which    Ri  ^  i?  -f 


LOCATION.  79 

Second  Method.— J.,  B,  aud  E  lie  on  the  same  straight  line, 
since  /  is  the  same  for  both  curves.  lu  triangle  BOE  angle 
EBG  =  \I,  aud 


BE  =    .       --  =  7)  cosec  47. 
sm  4/  ^ 


From  Table  IX,      AB  =  h^-'^^ 


D 


AE  =  AB  -\-  BE  is  the  long  chord  for  curve  of  degree  Dr, 
therefore 


^  _L.c.^i.r 

U\  —  — 


AE 


If  desired,  R  may  be  found  by  (12')  or  Table  I. 
Third  Method. — Draw  FL  parallel  to  OiE\  then 


CF  =  ~—r  =  P  cosec  /. 
sm  / 


From  Table  IX,       ^(7  =  ^'  ^  ^ 


AF=  AC -\-  CF\  the  tangent  distance  for  second  curve  ;  hence 


D,= 


AF 


Remark. — If  transit  is  set  up  at  B,  it  will  be  well  to  set  E 
by  measurement  from  B,  to  serve  as  a  check  when  the  curve  is 
run  in  from  A. 


80        A    FIELD-MANUAL    FOR    RAILROAD    EXGIXEERS. 


Akticle  9.     Compound  Curves. 
A,  Location  Problems. 

117.  Given  Two  Unequal  Tangents,  their  Intersection-angle, 
and  One  Radius,  to  Find  the  Other  Radius  of  a  Compound 
Curve  uniting  Tangents. 

In  Fig.  42,  An=  T,  and  BH=  T^  are  the  known  tangents, 
AOi  =  i?,  the  known  radius.  BO^.  =  B.  and  the  angles /i  and  /a 
must  be  found  before  curve  can  be  located. 


Fig.  42. 

Extend  first  branch  to  F,  so  that  tangent  FL  is  parallel  to  BE. 

Draw  ^A'and  BG  perpendicular  to  FL  ;  draw  FB  and  extend 
to  ^.  it  will  pass  through  the  P.C.C.,  because  the  central  angles 
EO,  F  and  EO^B  are  equal.     Then 

To  =  AL  =  Ri  tan  ^1. 


In  triangle  LHK,  since  LH  —  T^  —  Ti , 

p  =  HE  =  BG  =  (To  -  T,)  sin  I. 
Now  in  triangle  BGFsLUgle  BFG  =  \h,  and 

l~-FG=T,^s~  T,, 


LOCATION".  81 

tan  1/2  =  ^ (62) 

Draw  0,3/ parallel  to  FL  ;  then 

{B.\  —  li'i)  sin  Ii=^l, 
whence 

/?3  —  i?i : — T  =  Bi  —  I  cosec  I2.     .     .    (63) 

sin  /, 

Had  It}  been  required,  the  equation  would  have  been 

Ri  =  R^  -}- 1  cosec  /a. 

Evidently,  I^  =  I  —  I^. 

In  the  field  the  points  E  and  5  may  be  located  by  running  in 
the  curve  from  A  as  starting-point,  or  run  the  chord 

AF='iR^^\nlI 

from  A,  and  at  i^  deflect  angle  AFB  =  i/—  \I^  =  |/,  ,  measure 
i^i?  ^  I  sec  ^/a  and  i?£:  =  2i?2  sin  Ih. 

Example.— A  2°  curve  has  the  P.O.  at  sta.  110,  T,  =  590  ft., 
^2  =  511  8  ft.,  7  =  30°  50'.     Locate  the  curve. 

By  Table  IX,  T,  =  1580/2  =  790  ft. 

By  formulas  above, 

«  z=  200  X  0.85866  =  171.73  ft, 
i?  =  200  X  0.51254  =  102.51  ft., 
^  =  790  +  171.73  -  511.8  =  449.93, 

■JQO  51 

tan  1/2  =  TTTTWrf  =  0.22778  =  tan  25°  40'. 
449.97 

Then  I,  =  30°  50'  -  25°  40'  =  5°  10'. 

440  07 
R,  =  2864.93  -  ^---  =  1833  feet. 
.43313 


82        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

By  Table  1  this  is  seen  to  be  ihe  radius  of  a  3°  7^'  curve. 

Tlie  length  of  first,  brauch  is  258.3  feet,  and  of  the  second  821.3 
feet;  hence  the  P.C.C.  falls  at  112  +  58.3,  while  the  P.T.  is  at 
sta.  120  4-  79.6. 

118.  Given  the  Long  Chord  from  P.O.  to  P.T.  of  a  Com- 
pound Curve,  the  Angles  it  makes  with  the  Tangents  and 
One  Radius,  to  Find  the  Other  Radius  and  the  Central  Angles, 

In  Fig.  42  AB  is  known,  as  also  the  angles  HAB  =  a  and 
HBA  =  b.  Two  angles  and  one  side  of  the  triangle  HAB  are 
known,  and  the  sides  HA  =  Ti  and  HB  =  T^  may  be  found, 
after  which  the  solution  is  the  same  as  in  the  last  problem. 

A  solution  may  be  reached  in  a  different  manner.  I  =  a  -\-  b, 
HAF  =  U  =  h{a  +  b),  and  BAF  =  i{a -{- b)  -  a  =  l(b  -  a), 
AF  =  2Ri  sin  ^I.  In  triangle  BAF  two  sides  and  the  included 
angle  are  now  known,  so  BF  and  angle  BFA  may  be  found; 
GFB  =s  ^h  =  K  -  ^^-4. 

Then  EF  =  2R,  sin  ^h  , 

and  EB  =  EF  —  BF  hecomes  known. 

Then  EB  =  2B^  sin  ^7,  =  2i?i  sin  ^/j  -  BF, 

BF 
whence  ij,  =  if,  _  ___ (64) 

Evidently  /,  =  I  —  I^ 

119.  Given  the  Radii  and  Central  Angles  of  a  Compound 
Curve  to  Find  the  Tangent  Lengths,  the  Long  Chord  from 
PC.  to  P. 2.,  and  the  Angles  it  makes  with  Tangents. 

In  Fig.  43  draw  AE  and  BE  from  the 
PC.  and  P.T.  to  the  P.C.C,  then 
calculate  AE  and  BE  by  (16)  or  by 
Table  IX.  In  triangle  AEB  angle 
AEB  ==180  1(7,  +  7^).  Two  sides 
and  the  included  angle  being  known, 
the  triangle  AEB  may  be  solved  for 
AB  and  the  angles  ABE  and  BAE; 
then 

BAF-  BAE+^I,, 
Fig.  43.  j^^jr  ^  ^^^  ^  1 /^ 

The  angle  AFB  of  triangle  ABF  now  becomes  known  and,  as 


LOCATION.  .  83 

AB  is  kuown,  the  sides  yli*'' =  2\  imd  BF=  T^  may  be  com- 
puted. 

120.  Given  the  Long  Chord  from  P.C.  to  P.T,  of  a  Com- 
pound Curve  and  the  Angles  it  makes  with  Tangents  to 
Find  the  Radii  when  the  Common  Tangent  is  Parallel  to  Long 
Chord. 

In  Fig.  43  let  Gil  he  parallel  to  AB,  and  OAB  =  a,  HBA  =  b 
known.     Then 

BAE  =  BAG  =  GEA  =  \a, 
and  ABE  =  EBB  =  HEB  =  |6. 

Also,  AEB  =  180°  -  i{a  +  b). 

In  triangle  AEB,  remembering  that 

sin  [180  -  i{a  +  b)]=  sin  i{a  +  b), 

sm  l{a  4-  o) 
and 

BE  =  —  ^  ^^°  ** 


sin  i(a  +  b)' 


Since  AOxE  =  a  and  EO-^B  —  b,  the  radii  Ri   and  i?2  may  be 
found  from  formula  (16),  or  (16a). 


lAE                  AB  sin  ^b 

.    .     .     (65) 

~~  sin  ia  ~  2  sin  la  .  sin  |(a  +  6)' 

IBE                AB  sin  |a 

.     .     .     (66) 

Example. — Required  R\  and  R^ ,  or  Di  and  Da ,  when  AB  = 

900  feet,  a  =  13%  b  =  15°. 

By  (65),  Rt  =  2407.0  ft. 

By  (66),  Ri  =  1543.7  ft. 

From  Table  I,     /).  =  2°  22'  50"     and     D^  =  3°  42'  44". 


84        A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


B.    Obstacles. 

121.  To  Locate  a  Point  on  i-ne  Second  Branch  of  a  Com- 
pound Curve  when  the  F.C.C.   is  Inaccessible. 

Ordinarily  the  second  branch  is  located  by  setting  transit  at  the 
P.C.C.  and  running  the  curve  from  that  point.  An  obstacle  on 
either  curve  may  then  be  passed  by  the  methods  given  for  simple 
curves. 

*  When  the  P.C.C.  is  inaccessible, 

aQ  locate  the  first  branch  from  theP.C. 

and  the  second  branch  from  the 
P.T.,  if  this  latter  point  is  known. 
When  this  is  not  the  case  proceed 
by  one  of  the  following  methods: 
First.  By  means  of  a  long 
cliord. 

In  Fig.  44  let  ^  be  the  P.C.C, 
A  some  known  point  on  first 
branch,  EF  a  tangent  at  E,  and 
AB  parallel  to  FE.  The  station 
numbers  of  A  and  E  being 
know^n,  the  arc  AE  and  angle  a 


Fig.  44. 


are  readily  found  ;  then 

EL  =  2?2  vers  b  =  Ri  vers  a. 


whence 


next, 


vers  b  — 


lit  vers  a 
Hi 


(67) 


AB  =  El  sin  a  -\-  Bu  sin  b (68) 

Deflect  FAB  =  a  from  tangent  at  A  ;  measure  out  AB  ;  set  the 
transit  at  B  and  locate  the  second  branch. 

By  Table  IX.  —Take  the  mid-ordinate  in  table  for  an  inter- 
section-angle 2a  ;  then 

i¥,  ^  2a 


EL  = 


B 


Then  EL  X  D^  is  the  mid-ordinate  for  a  1°  curve  having 
I  =  2b,  from  which  b  becomes  known.  From  the  table  now  find 
AL  and  LB,  the  half-chords  for  angles  2a  and  2b,  and  proceed  as 
before. 

Second  Method. — By  means  of  tangents. 

From  Fiix.  44,         AF  =  FE  =  11,  tan  la. 


LOCATION. 


85 


Set  transit  at  F,  deflect  OFE  =  a,  and  by  some  indirect  method 
measure  to  an  accessible  point  H. 


and 


EH  =  FH  -  FE, 


EH 


tan  \h  =  -^-,  from  formula  (14). 


Angle  b  is  now  known  and  equals  GIIE,  wbicb  deflect  from 
EH;  then  measure  JIB  =  EH,  and  with  transit  at  B  locate  the 
second  branch  of  curve. 

Or  by  Table  IX.— Find  AF  =  FE,  the  tangent  distance  for 
I  =  a;  then  having  j^Zf  measured,  take  Ti  =  EH  X  -O2  and  find 
the  corresponding  angle,  which  equals  b  ;  then  proceed  to  locate 
curve  as  above. 

Example.— Let  A  be  at  sta.  126,  P.  C.  C.  at  128  +  25;  the  degree 
of  first  branch  4°,  and  of  second  6°. 

By  the  first  method  EL  =  17.635  for  a  =  9°,  and  6  =  11°  2', 
nearly.  AL  =  224.1  ft.,  BL  =  182.75  ft.,  and  therefore  AB  = 
406.85  ft.  Angle  b  =  11°  2'  corresponds  to  183.9  ft.  around  6° 
curve;  hence  the  P.T.  number  is  130  +  08.9. 

By  the  second  method  AF  =  112.74  ft.  Suppose  FH  =  264  ft., 
then  EH=  151.26  ft.,  which  multiplied  by  6  gives  907.56  ft., 
corresponding  to  /  =  18°.  The  arc  EB  is  now  800  ft.,  making 
B  fall  at  st*.  131  +  25. 


C.     Change  of  Location. 

122.  Having  a  Simple  Curve  Located  to  Find  the  P.C.C.  so 
that   a  Curve  of  Given  Radius  shall  connect  with  a  Given 

Tangent    Parallel     to     Tangent     to 
Located  Curve. 

Let  NAB,  Fig.  45,  be  the  located 
curve,  HF  the  tangent  in  which  the 
second  branch  must  end.  The  dis- 
tance BG  =  p  between  tangents  is 
known  from  measurement.  If  angle 
a  can  be  found,  the  arc  BA  becomes 
known  and  the  point  ^1  can  be  located 
from  B.  Draw  O^L  from  the  center 
of  second  branch  perpendicular  to 
Fig.  45.  0.J5.      In    triangle    O.O-.L,    0,0.^- 

Ri  —  Ri,  and  OiL  -  Ri  —  {R-i  +  p) ;  therefore 


86        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


cos  a  =  ^ r; —  =  1 r, 77  •      •     •     \^^) 


Ri  —  xij 


Jrii  —  J?2 


Then  a  divided  by  Z),  gives  arc  BA. 

If  desired,  BR  may  be  found  from  the  right  triangle  BEG,  in 
•which  the  side  BG  =  p  and  angle  GHB  =  Ui  are  known— 
A,  E,  and  B  lying  in  the  same  straight  line  ;  then 


BE  = 


V 


=  p  cosec  ^a. 


(70) 


sin  ^a 

Or  BA  and  EA  may  be  found  from  Table  IX,  after  which 
BE  =  BA  -  EA. 

Example.— A  3°  curve  ends  in  a  tangent  at  sta.  160  -f  50, 
35  ft.  outside  of  desired  tangent.  Find  the  point  of  compound- 
ing with  a  4°  50'  curve. 

From  Table  I,  R  for  3°  curve  equals  1910.08  ft.,  and  for 
4°  50'  curve  1185.78  ft. 


Then,  by  (69),    cos  a  =  1  -  ^ 


35 


'243 


0.95168. 


From  table  of  cosines  angle  a  is  found  to  be  17°  53'.  Dividing 
this  by  3  gives  5.961  stations  for  the  arc  BA.  Hence  the  P.  CO. 
number  is  160.50  -  5.961  =  sta.154  +  53.9,  and  the  new  P.T.  is 
at  sta.  158  +  23.9. 

123.  Given  a  Located  Compound  Curve  ending  in  a 
Tangent  Parallel  to,  and  a  Given  Distance  from,  a  Tangent 
in  which  the  Curve  is  required  to  end.  To  Find  the  Neces- 
sary Change  in  P.  C.  C. 

First  Qa^^.— Terminal  branch  hamng  shorter  radius. 

In  Fio;.  46  let  ABC  be  the  located 
curve,  AEF  the  one  required  ;  angle 
BOiC  =  a  known,  and  also  J/JV  =  p. 

If  angle  EOM  =  b  can  be  found,  the 
angle  of  retreat  from  B  \.o  E  will  equal 
b  —  a. 

Draw  0/A'  and  0,Z  perpendicular 
to  ON,  which  is  parallel  to  OiC. 


Fig.  46. 


Then     OK  =  (R-  Ri)  cos  b, 
OL  —  {R  —  Rx)  cos  a. 


LOCATION. 


87 


Now  LM=B,  -  KL  =  Ii,  -  MN,  from  which  KL  =  MN  =  p. 
Hence 

{R  -  Rx)  cos  6  =  (i?  -  Rx)  cos  a  -  p. 
From  which 

cos  b  —  cos  a  —  -=5 5r ^  ^^ ) 

i\  —  ill 

Divide  &  -  a  by  D,  the  curvature  of  first  branch,  and  move 
back  that  number  of  staiions  from  B  to  the  new  P.G.V.  at  E. 

Join  0.0,' \  evidently  FC  =  0.0/,  and  angle  KO.'O,  =  CFG  ; 
00/Ox  =  90^  -  1(6  -  «),  00x'K=  90°  -  i.     Hence 

(7FGf  =  KO.'Ox  =  [90  -  K^  -  a)]  -  (90  -  &)  =  i(6  +  «)•     (73) 
From  triangle  CGF, 

FC  =  -7— r-r-7— ^  ^  ^  cos^^  K^  +  «)•     •    •    C^^) 

sm  ^{b  -\-  a) 

Or,  from  triangle  Od'Oi , 

FC  =  Ox'Oi  =  2{R  -  Rx)  sin  K^  -  a). 

Had  ^i?F  been  the  original  curve,  b  would  have  been  known 
and  a  required. 

From  (71),  cos  a  =  cos  6  -f  ^-^-^ C^^) 

Ci^* and  angle  Ci^Jf  are  given  by  formulas  (73)  and  (72). 

Example.— A  2°  curve  compounds  with  a  4°  curve  at  sta. 
82  -I-  30;  a  =  20'  30',  p  =  40  feet.  Find  number  of  new  P.C.C. 
and  distance  between  P.T.s. 

40 
From  (71),   cos  b  =  0.93667  -  2864.9  -  1432.7  "  ^•^^^'^^• 

This  yields  b  =  24°  20',  and  &  -  a  =  3°  50'. 

3  833 
The  change    in  P.C.C.  is  -^  =  1.917  stations;   the  P.C.C. 

number  is  therefore  82  30  -  1.917  =  sta.  80  +  38.3. 


88        A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


By  (72),  CFO  =  ^(24°  20'  +  20°  30')  =  22^  25'. 

By  (73),  i?'(7  =  40  X  2.62234  =  104.9  feet. 

Second  Case. — The    terminal   branch  having   longer  radius. 

Let  CAB,  Fig.  47,  be  the  located 
curve  with  P.C.C.  at  A,  aud  let 
FK  be  the  tangent  in  which  the 
curve  is  required  to  end. 

The  distance  BK  =  p,  the  radii 
OA  =  R,  OiA  =  Bi ,  and  angle 
AOiB  =  a  being  known,  it  will 
be  sufficient  to  find  angle  EOi'F 
in  order  to  get  the  angle  of  ad- 
vance, AOE  =  a  —  b.  Draw  OL 
and  OiN  perpendicular  to  O/F 
Fi«.  47.  and    OiB.       From    the    triangles 

0,'Oilf  aud  O.OZ, 

(R,  -  R)  cos  b  =  Ox'N-\-  {Ri  -  R)  cos  a. 
But  Oi'N  =  KB  =  p;  therefore 

(7?i  —  R)  cosb  =p-^  (Ri  —  R)  cos  a. 


o'^^ 

k\ 

-1/ 

\ 

Oi 

L 

K 
H 

^p-^ 

B 


Whence 


cos  b  =  cos  a  -f- 


P 


Rx  -  R' 


Then  — jr-  will  be  length  of  curve  from  A  to  E, 


(75) 


Angle  KFB  =  N0,0,'  =  00,0,'  -  N0,0. 
But   00,0,'  "  90°  -  i(a  -  b)    and    N0,0  =  90  -  o. 

•.     KFB  =  [90°  -  K«  -  *)]  -  [90  -  a]  =  K«  +  *)• 
From  triangle  KFB, 


FB  = 


V 


sin  \{a  -\-  b) 

Or,  from  triangle  0,00,',  since  0,0,'  -  FB, 
FB  =  2(/?.   -  R)  sin  4(a  -  b). 


=  ^  .  cosec  \{a  -\-b).   .     .    .     (76) 


LOCATION. 


89 


If  AEFhn^  been  tbe  located  curve,  b  would  have  been  given 
and  a  required.     From  forumla  (75), 


cos  a  =  cos  b  — 


P 


Hi  -  M' 


(77) 


Example. — A  5°  curve  compounds  at  sta.  60  with  a  2"  curve, 
and  the  P.T.  is  at  sta.  80.  What  will  be  the  number  of  F.C.C. 
if  the  P.7\  fall  in  a  tangent  81  feet  inside  of  terminal  tangent? 
Here  a  =  40°. 


81 
By  (75),        cos  b  =  0.76604  +  -—  =  0.81316. 


Hence  b  —  35°  36'  and  a  -  b  =  4°  24',  corresponding  to  220 
feet  around  the  2°  curve.  The  number^  of  the  new  P.C.C.  is 
therefore  62  +  20, 

angle  KFB  =  1{W  0'  +  35°  36')  =  37°  48', 


and 


FB=S\y<  1.63157  =  132.16  feet. 


124.  Given  a  Located  Compound  Curve  to  Find  Necessary- 
Change  in  r.C.C.  and  Radius  of  Second  Branch  to  make  the 
P.T.  fall  in  a  Tangent  Parallel  to  First  Terminal  Tangent 
and  in  a  Point  on  the  Same  Radial  Line. 

First  Case. — Second  brunch  having  shorter  radius. 

In  Fig.  48, OB^R, O.B^Ry  angle 
a  and  HG  =  p  are  known.  02E=R<i 
and  angle  b  must  be  found  ;    then 

— — -  =  BE  will  be  the  change  in 

P.G.G. 

Produce  first  branch  to  K,  where 
OK  is  parallel  to  0,  G.  Since  BOK 
=  B0^  G,  B,  K,  and  G  lie  in  the  same 
straight  line;  and  since  EO^F  ~  q 
EOK,  E,  F,  and  K  lie  in  the  same 
straiiiht  line.     Tlierefore    • 


KCG=la,     and     KFHz=\b. 


Fig.  48. 


90        A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


From  triangles  KFH  Siud  KCG, 


HK       OK       GH 


But 


tani6  =  ^^  =  ^+^^=tauia  + 


FH=  OxL  =  (R-Rx)  sin  a. 


FH 


.'.     tan  ^b  =  tau  ha  + 


P 


{R-  7?,)  siu  a 


From  triangles  OOiL  and  OO-^M, 


(R  -  Ri)  siu  b  =  {R-  Ri)  sin  a. 


When 


R^  =  R-{R  -  R,) 


sm  a 


sin  6 


•  • 


(78) 


(79, 


Had   AEF  been   the   first  curve  located,  b  and  Rt  would  be 
kuown,  a  aud  Ri  required. 

From  the  figure,  reasoning  as  before, 


tan  ^a  =  tan  ^b  — 


P 


{R  -  i?j)  sin  b' 


(80) 


and 


Rx  =  R~  {R-  R,) 


sin  b 


sm  a 


.     .     (81) 


Second  Case. — Second  branch  having  longer  radius. 

.A 


Fig.  49. 


In  Fig.  49  let  AB  be  the  located  curve,  ^i^the  curve  required, 
OA  =  R,  O^A  =-  R,,  O^E  =  7?2.  FB  =  p. 


LOCATION.  91 

R,  aud  angle  b  are  wanted,  angle  a  being  known. 
We  can  show,  as  in  first  case,  that 

HFK  =  lb,     HBL  =  \a, 

OM  =  KF  =  LB  =  {Bx  -  B)  sin  a; 
and  hence 

^^^^-    -  FK~  BL       BL' 

Or  inserting  values, 

tan  lb  =  tan  ia  —  -^ ^^--. (82) 

^  ^         {Bi  —  B)  sin  a 

Angle  b  now  becomes  known  and  — ^— -  =  AE  in  chains,  which 

is  the  change  in  position  of  P.C.C. 
From  triangles  OOiM  and  OOiM, 

(i?a  -  B)  sin  b  =  {Bi-  B)  sin  a 


.-.  i?a  =  7?H-(i?i  -i?f4^ (83) 

sm  b 


Had  the  new  tangent  fallen  outside  the  old  one,  we  should  have 
had 

t^n^a  =  i^n^b+—-^y^^^,      .    .     .     (84) 


and 


H,  =  B -\~  {B,  -  Bf^.    .....    (85) 

sm  a 


125.  Having  a  Located  Compound  Curve,  to  Find  the 
Change  in  P. CO.  and  Radius  of  Second  Branch  in  order  to 
Cause  P.  T.  to  Fall  at  a  New  Point  in  Terminal  Tangent. 

First  Case.  —  Second  bnoich  lutclufj  shorter  radius. 


93        A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

In  Fig.  50  let  NAB  be  the  located  curve,  and  C  the  point  where 
r.T.  is  required  to  fall.  Let  BC  =  k,  OA  =  R,  0,B  =  R,,  and 
angle  Oi OH  =  ahe  known;  angle  h  and  R^  are  required. 


Extend  first  branch  to  F,  making  OF  parallel  to  OiB.  A,  B, 
and  i^lie  on  a  straight  line,  for  angles  J. 0,5 and  AOFsive  equal; 
likewise  E,  G,  and  F  lie  on  the  same  straight  line. 

From  triangles  GBFand  QCF, 


^  ^.       OB      CB  ^  ^  k 

cot  io  =  ^^TP,  —  -T^n^  =  cot  la  — 


GF      OF 


OF 


But      OF  =  EM  =  {R-  Ri){l  -  cos  a)  =  {R  -  i?,)  vers  a. 


cot  hb  =  cot  |a  —  -^ ^- . 

^        (R—  Ri)  vers  a 


(86) 


From  triangles  OOiE  and  OO^L,  since  OiP  =  k, 
{R  -  /e,)  sin  b  =  {R-  Ri)  sin  a  -  k. 


Whence 


ff,  =  R-\- 


k  —  {R  —  Ri)  sin  a 

sin  b 


(87) 


Then  b  —  a  divided  by  D  gives  arc  AF     With  radius  R^  locate 
the  curve  EC  from  C  or  E. 


LOCATION. 


93 


Had  NEC  been  the  located  curve,  R,  R2 ,  and  b  would  have 
been  known,  A',  and  a  required.     In  this  case 


k 


cot  la  =  cot  lb  -  _--^^-— -^^, 


(88) 


E,  =  E- 


k  '{-  {R—  R^)smb 


•  • 


sin  a 


.     (89) 


Second  Cask.  — Terminal  branch  having  longer  radius 

In  Fig.  51  let  NAB  be  the  located  and  iV^^Cthe  required  curve. 


Fig.  51. 


Let  GB  =  kha  known.     Then,  as  in  the  first  case, 


cot^&=  -^--7 


GC      GB        k 


FG       FG      FG' 


Hi 

.*.     cot  \b  =  cot  la -=- zT- , 

^  ^  (A'l  —  R)  vers  a 


•  • 


90) 


and 


(Rj  -  R)  sin  a=  {R2-  R)  sin  b-\-k; 


whence 


S,=  B+   '■^■-^.)7«-^     .     .     (91) 

sind 


94        A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


Had  NEC  beeu  located  and  NAB  required,  the  equations  would 
have  been 


and 


cot  \a  =  cot  \h  -\- 


b   1 

k 

'  {R. 

— 

B) 

vers  6 

(R,- 

B) 

sin 

b-\-k 

sin  a 


,     .     .     (92) 
.     (93) 


•  • 


In  either  of  these  two  cases  if  k  is  unknown  and  the  new  radius 
given  or  assumed,  the  desired  angle  and  the  value  of  k  may  be 
found  from  the  foregoing  equations.  Or,  knowing  the  new  angle, 
the  new  radius  and  value  of  k  may  be  found  from  the  same 
equations. 

126.  To  Replace  a  Curve  of  Given  Radius,  which  unites 
Two  Tangents  with  Known  Intersection-angle,  by  a  Three- 
centered   Compound   Curve. 

In  Fig.  52    let  OA  =   R   he   the    radius  of    located   curve, 


O2C  =  OiA  =  Ri  the  radius  of  terminal  portions  of  the  three- 
centered  curve,  and  the  other  notation  as  shown  in  the  figure. 

Draw  OiOi',  and  draw  i^Oi/ perpendicular  thereto.     From  tri- 
angles O^Oi^ff  and  OjOff, 

O^H  =  (R,  -  R,)  sin  ^/,  =  {R^  -  R)  sin  ^/.  .     .     .     (a) 

Suppose  i?a  and  Ri  to  be  assumed  ;  then  equation  (a)  yields 


sin  4/1  = 


x?2  —  R 

Ri  —  Ri 


sin  i  J. 


(94) 


LOCATION'.  95 

Then  AfhET^CO^G  =  l{I-Ix).      .     .     .     (95) 

Suppose  AO-i'E,  COtO,  and  ^2  to  have  been  assumed.  From 
(95)  find  /i  ;  then,  from  equation  (a), 

sin  ~I 

Br  =  R,  -  (li,  -  B)~^^ (96) 

sm  ^/i 

Example. — Given  a  4°  curve,  7=38°,  and  the  terminal 
branches  composed  of  a  2°  curve  for  two  stations,  to  find  Jii  and 
Di  for  the  central  portion. 

Here  /,  =  38°  -  2(2  X  2)°  =  30°. 

From  Table  I,  R^  =  2865  ft.,     R=  1432.7  ft. 

Whence  R^-R  =  1432.3  ft. 

Log  1432.3       =  3.15603 
"     sin  19°  0'  =  9.51264 


2.66867 
"    sin  15"  0'  =  9.41300 


.-.     log  1801.7  =  3.25567 

Therefore  R,  =  2865  -  1801.7  =  1063.3  ft.,  and,  by  Table  I, 
Bi  —  5°  23  .4,  nearly  enough. 

127.  To  Substitute  a  Curve  of  Given  Radius  for  a  Tangent 
uniting  Two  Curves. 

In  Fig.  53  let  the  tangent  BC  =  i,  OB  =  R,  0^0  =R^,  and 
0,A  =  Ri  be  known. 

Angles  a,  h,  and  c  must  be  found  in  order  to  substitute  curve 
AE  for  the  system  ABCE. 

Draw  OF  parallel  to  BC,  then  O^F  =  R^  —  R,  and,  from  triangle 
00,F, 

^^d  =  ^-4^. (97) 

t 


00,  =  -7-^  =  t .  cosec  (I  =  i^{R,  -  Rf  -f- 1^.      .     (98) 
sm  o 


9G        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


Now  in  triangle  00\0o  three  sides  are  known  and  the  angles 
c  and  e  may  be  computed.    Thus  if  s  is  the  half-sum  of  the  sides, 


cos  ic 


/ 


s{s  -  00,) 
00^  X  OaOi" 


Angle  e  may  be  found  in  like  manner,  then  b  =  180"—  (e  -j-  d), 
and  a  =  c  —  b. 

Points  A  and  ^may  now  be  located  and  the  curve  traced. 

Example. — A  3°  and  a  5°  curve  are  united  by  a  tangent  500 
feet  long.     Replace  by  a  2°  curve. 

Here         R,  -  B  =  1910  -  1146  =  764  feet. 


By  (97),        tan  d 


500 
764 


=  0.65444  =  tan  33°  12' 


By  (98). 


00,  =  913.1  feet. 


In   triangle    00,0^,  00,  =913.1,    0,0^ 
1718.7  feet.     Solving  for  e  and  c, 


=  954.9,  and    OO3  = 


e  -  133°  36',     c  =  23°  0'.     Then     b  =  13°  12',     a  =  »°  56'. 


Article  10.     Track  Problems. 

128,  Reversed  Curves  should  never  be  employed  on  main 
lines  because  of  the  shock  due  to  sudden  reversal  of  curvature 
and  superelevation  of  outside  rail.  A  short  tangent  should  be 
interposed  between  the  two  curves,  which  ma}'  ordinarily  be 
done  by  changing  the  end-points  of  the  curve,  or  slightly  altering 
the  radius.     If,  however,  transition  curves  are  emi^loyed  to  ease 


LOCATION. 


or 


off  both  curves,  there  would  seem  to  be  no  objection  lo  the  use  of 
curves  of  contrary  flexure,  provided  the  track  may  be  kept 
always  in  perfect  condition.  In  yards,  crossovers,  and  where 
connection  is  made  with  existing-  track,  reversed  curves  may  be 
employed,  and  are  often  imperative. 

129.  Having  a  Located  Curve  Intersected  by  a  Straight 
Line,  to  Connect  them  by  Another  Curve. 

Either  the  radius  of  the  joining  curve  may  be  given,  or  else  the 
point  on  first  curve  at  which  the  junction  must  be  made.  The 
angle  between  a  tangent  to  located  curve  at  the  point  of  meeting 
and  the  straight  line  must  be  measured.  Four  possible  cases 
occur. 

First  Ca&^.— Joining  curve  tangent  to  located  curve  internally 
and  on  same  side  of  cutting  line  as  center. 

In  Fig.  54  let  GF  be  joining  curve,  with  center  Oi  and  radius 
Ri.  Let  radius  of  located  curve  OF  =  B.  Draw  OiG  and  Oil 
perpendicular  to  the  cutting  line  produced,  and  OiK  parallel  to 
AH.     If  Bi  is  known,  we  must  determine  angle  b,  a  having  been 


Fig.  54. 
measured ;  then  b  —  a  gives  the  length  of  arc  from  A  to  ii^  where 
the  P.O. C.  is  to  be  located.     In  the  triangle  KOOi  we  have 

OK  =  OH-  Ri    and     00,  =  R  -  R,. 

R  cos  a  —  Ri 


Then 


cos  b  = 
b  —  a 


D 


R  —  Ri 

=  arc  AF. 


(99) 


Had  i^been  given,  we  should  have  b  =  a-\-AOF,  and,  from  (99), 
n    __  R  (cos  a  —  COS  b)  _R  (cos  a  —  cos  b) 


1  —  cos  b 


vers  b 


98        A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


Example. — A  1°  curve  is  cut  by  a  langeut  that  makes  an  angle 
of  64"  32'  with  tangent  to  curve.     Unite  by  means  of  a  4°  curve. 

By  (99),  cos  b  =  0.24000  =  cos  76°  07',  and  therefore  b  -  a  = 
11°  35',  making  AF,  of  figure,  11.58  stations. 

Second  Case.  —  Joining  curve  tangent  internally  to  located 
curve  but  on  opposite  side  of  cutting  line  from  center  of  located 
curve. 

In  Fig.  54  let  arc  ME,  with  center  O2  and  radius  i?2,  be  the 
joining  curve.     From  the  figure, 


cos  d  = 


R  cos  a  +  i?2 


(101) 


Then  arc  AE  =  a  -  d  divided  by  B,  and  c  =  180°  -  d. 
Had  the  point  j&  been  given   and  i^a  required,  it  would  have 
been,  from  (101) 

i?(cos  d  —  cos  a) 


2?2   = 


(102) 


1  -\-  cos  d 

Example. — Take  the  same  example  as  in  first  case.     Here, 

By  (101),         cos  d  =  0.9068  =  cos  24°  56'. 

Then  64°  32'  -  24°  56'  =  39"  36', 

equivalent  to  39.600  stations  around  curve  from  A  to  E. 

Third  Case. — Joining  curve  tangent  externally  to  located  curve, 
with  center  on  same  side  of  catting  line. 


L 

N. 

/ 

/ 

kO. 

G 

o^X^vM/ 

c 

-> 

R, 

F 
E 

I            />-- 

. 

,H          C 

\ 

\  1          ' 

^"h. 

/     • 

'^^.---^'r'""^ 

" 

3, 

0 

1 

Fig.  55. 


In  Fig.  55  let  arc  5(7,with  center  0,  and  radins  7?, ,  be  the  join- 
ing curve.  Draw  O^E  parallel  to  CF,  aud  OiCand  OF  perpen- 
dicuhir  thereto. 


LOCATION.  99 

From  the  figure, 

(R  +  -Ki)  cos  h  =  R  cos  a  —  jRi; 

-       i?cos  a  —  i?i  ,^__. 

.'.  cos6  =  — -— -•^— - (103) 

Then  d  =  180  —  h,  and  J^05  =  b  —  a.  The  curve  may  now  be 
traced  on  the  ground. 

If  ^C  is  wanted,  we  have  AC  =  (i?  +  ^0  sin  &  —  i?  sin  a. 

If  the  point -B  is  fixed  and  Ei  required,  there  results,  from  (103), 

It  (CO,  a- cos  b) _ 

1  +  cos  0 

Example. — Take  the  example  given  for  the  first  and  second 
cases 
By  (103), 

,       5730x0.43-1432.5       ^  .  .  .  _,„.,, 

""^  ^  =       5730  +  1432.5        =  ^'^^^  =  ^^  ^^    ^^ 

6  -  a  =  81°  44'  -  64°  32'  =  17°  12',  equivalent  to  17.2 

stations  on  located  curve  from  A  to  B.  Angle  d  —  180"  —  81"  44' 
=  98°  16',  equivalent  to  24,567  stations  from  B  to  C  on  the 
4°  curve. 

Fourth  Case, — Joining  curve  tangent  externally  to  located  curve, 
with  center  on  opposite  side  of  cutting  line. 

Let  O2,  Fig  55,  be  center  of  joining  curve,  R2  its  radius. 
From  the  figure, 

{R  -\-  R2)  cos  c  =  R  cos  a  +  i?2. 

■ 

R  cos  a  -{-  R2  ,^^^, 

.  ••  cos  c  =  — -g-^-^-- (105) 

If  Jf  is  fixed  and  R2  required,  (105)  yields 

„    _  i?(C0S  C  —  cos  a)         R(C0S  C  —  cos  a)  /ir\a\ 

Hi  —  :- =   — ; .  .       (10b) 

1  —  cos  c  versm  c 

Example. — Take  same  example  as  in  preceding  cases. 
By  (105),  cos  c  =  0.54403  =  cos  57°  02'. 

Then  a  -  c  =  64°  32'  -  57°  2'  =  7°  30', 


100     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


calliug  for  a  distance   of  7.50  stations  from  A  to  M  around 


1°  curve. 


From  MtoHon  4°  curve  is  14.258  stations. 


130.  To  Locate  a  Y 

A  Y  is  made  up  of  a  system  of  tracks  so  arranged  as  to  admit 
of  turning  an  entire  train.  Tliree  of  the  most  used  arrangements 
are  given  belovs^. 

First  Case. — One  branch  of  T a  straight  line. 

This  is  only  the  special  case  of  the  last  problem  in  which  the 
cutting  line  becomes  tangent  to  both  curves.    In  Fig.  56,  if  any 


A                           F 

C 

J 

^                   "^"v.^                     ''\ 

^0^^"^      '  '' 

^■^\ /'  \ 

^  V                        ^^ 

E 

R             /           N 

x\l. ; 2,'^ 

\? 

"^1 

0 

'^'''"'^ 

Fig.  56. 


one  of  the  points  A,  B,  or  G  is  given,  the  others  may  be  located 
by  landing  the  angles  c  and  b.  Draw  OiE  parallel  to  CA  ;  then 
in  triangle  OOiE 

{B  4-  El)  cosb  =  R-  Ri. 


cos  b  = 


R  —  Ri 
R-\-  Ri 


(107) 


This  follows  at  once  from  (103)  by  making  angle  a  =  0.  Then 
angle  c  =  180  —  b.  If  AB  were  a  located  curve  and  the  point 
^glvens  formula  (107)  would  furnish  us  a  value  for  i?i. 

Another  solution  is  to  produce  the  tangent  at  B  to  cut  AC  at  F; 
then  AF  =  FG  =  BF.  Join  i^with  0  and  0,  ;  it  can  easily  be 
seen  that  angle  OFOt  =  90°,  and,  by  geometry. 


Therefore 


tan 


BF=  \/RX  Ri. 
.       BF_    /Rr 


and 


^      ^        BF       /R 


(108) 
(109) 

(110) 


LOCATION. 


101 


Example.— Let  AB  be  a  3°  curve,  BG&  6°  curve,  the  poiut  A 
at  station  180. 
By  (107), 

cos  b  =  !o!n  !  7  lli^A  =  0-^317  =  cos  70°  32'. 
1910.1  4-  955.4 


The  nutQber  of  B  is  180  +  23.511  =  203  +  51.1.     Angle  c  = 
109°  28',  equivalent  to  18.244  stations  on  the  6°  curve. 

Second  Case. — 77ie  tliree  branches  curved  and  convex  towards 
each  other. 

Given  the  three  radii  and  any 
one  of  the  points  A,  B,  or  G, 
Fig.  57,  we  have  only  to  find  the 
angles  at  the  center,  then  divide 
these  angles  by  the  degrees  of 
the  respective  curves  to  get  their 
lengths  and  locate  the  three 
branches. 

In   the  triangle   OOiOi,  letting  ^ 
OOi  =  I,     OiOi  =  m,     OOi  =  n, 

we  shall  have,  by  trigonometry, 


Fia.  57. 


cos 


f      m  .  n        r    I 


(i?  H-  -Ri  +  Bi)B^ 
{R-\- B^){Rx -^  B,) 


(111) 


Angles  b  and  c  may  be  found  in  like  manner. 

The  angles  may  be  found  otherwise  by  letting  fall  a  perpen- 
dicular from  one  vertex  upon  the  opposite  side,  as  OE  perpen- 
dicular to  Oi  Oi.     Then  from  the  relation 

OiOa :  0x0  -f  00a  =  OOi  -  00^ :  OiE  -  O^E 


determine  O^E  and  OiE;  then  the  right  triangles  0^0 E  and 
OiOE  yield  values  of  cosine  a  and  cosine  c,  after  which  b  may 
readily  be  obtained. 

Third  Case. — One  branch  concave  to  the  otJier  two. 

In  Fig.  58  the  triangle  00,  O2  may  be  solved  for  the  angles  at 
0,  0) .  and  O-i ;  for  if  the  radii  are  given,  the  sides  OOj  =  B  —  Bi, 
OOi  =  .K  —  Ri,  and  OiOt  =  Bi  -\~  Bi  are  known  and  the  solution 


102     A   FIELD-MANUAL   FOR  RAILROAD   ENGIKEERS. 

is  the  same  as  for  second  ease.  Then  b  is  the  central  angle  for 
curve  AB,  a'  =  180  —  a,  the  central  angle  for  AG,  and  c'  = 
180  —  c,  the  central  angle  for  curve  BG. 


Example. — If  A  is  at  sta.   820  on  the  1°  curve  AB,  AG  an 
8°  curve,  connect  with  a  6°  curve  GB.     Here  we  have 

OaO  =  5730  -  717  =  5013,    0x0  =  5730  -  955  =  4775, 

and  OaOi  =  955  +  717  =  1672. 

Solving   this  triangle,   we   get    c  =  88°  20',    b  -  19°  28',    and 
a  =  72°  12'.      The  number    of  B  is   therefore   820  +  19.467  = 

839  +  46.7  ;  the  length  of  GBis  — -—  =  15.278  stations,    and 

107  8 
of  J.  (7  is       '     =  13.475  stations. 
8 

131.    To    Locate    a    Reversed     Curve    between    Parallel 
Tangents. 

First  Case. — Radii  equal. 

(a)  The  equal  radii  B  and  distance  p  between  tangents  known. 
In  Fig.  59  draw  0^  parallel  to  AG  to  meet  OiB  produced. 
From  triangle  OEOi, 

and  OE  =  2Ii&\na (113) 


LOCATION. 


103 


From  triangle  ABG, 


P 


AB  -  —^-~  -  p  cosec  iitt  =  \/0E^  -\-  p\       .     (114) 

Sill  "^Qf  , 


Fig.  59, 

(b)  AQ  and  p  known,  R  required. 

Here  AB  =  \^'AG-  +  p^  =  k.  Draw  OH  to  the  mid-point  of 
AC.  Triangles  J^O^  and  ABO  are  similar  and  AH  =  \k. 
Therefore 

K  -A. 
\k-p* 


whence 


R  = 


Ap 


(115) 


Example. — Connect  two  parallel  tracks,  30  ft.  c.  to  c.  by  a  T 
reversed  curve.     From  Table  1,  B  =  819  feet,  and,  by  (113), 


cos  a  =  \ 


30 

1638 


=  0.98167  =  cos  10'  59'. 


By  (113),     OE  =  1638  X  .19053  =  312.1  feet. 


By  (114),     AB  =  i/(313.1)'^  +  (30)^  =  313.5  feet. 


If  jo  =  30,  OE  =  312.1,    or  AB  =  313.5  had  been  given,  we 
should  have  had,  by  (115) 


R  =  1??M)!  =  819  feet. 

XfwU 


104     A   FIELDoiANUAL   FOR   RAILROAD    ENGINEERS. 

Second  Case. — Radii  unequal. 

(a)  Suppose  the  radii  R  =  OA  and  Ri  —  0,B  (Fig.  59)  to  be 
known.  "We  must  find  central  angle  a  and  AB  =  k.  From  the 
trjangle  00i£^ 

Then  AB  will  be  given  by  (114). 

{b)  Suppose  AB  =  k,  p  and  R  known,  to  find  Ri  and  angle  a. 

Triangle  ABG  yields 

sin  ia  =  -|- (117) 

OiLB  is  similar  to  A  OB.     Hence 

Ri  _  k 

LB~  p' 

But  AO  =  2R  sin  ^a,  and  LB  =  i{k  -  AC)  =  iCi.  Inserting 
this  value  of  LB  and  solving  for  Ri, 

«■  =  ^-   •     • ("8) 

From  similar  triangles, 

Ri_         Cr  ■ 

B       k-  Gi' 

Inserting  me  value  of  (7i  =  -^^  from  (118)  and  solving  for 
Ri ,  we  get 

R,  =  ~~-  R.    .    .    .    .    .    .    (119) 

2p 

Example.— ^5  =  300',    p  =  30',    R  =  819  ft.,  to  find  angle 

a  and  Rj. 

on 
By  (117),         sin  ^a  =  ^  =  0.10000  =  sin  5°  44'. 

Tliorefore  angle  a  =  11°  28'. 

Bv  (119\  Rx  =  ^^^  -  819  =  681  ft.,  an  8°  25'  curve. 


LOCATION". 


105 


132.  To  Connect  Two  Parallel  Tracks  by  a  Crossover  com- 
posed of  two  D°  Curves  with  a  Given  Length  of  Tangent 
between  Points  of  Contrary  Flexure. 

In  Fig.  60  let  AFOB  be  the  re- 
quired crossover,  FO  =  l,  EB=p, 
and  OA  =  OB  =  R  known ; 
angle  a  and  AE  =  x  are  re- 
quired. 

Draw   OM  parallel   to  AE  to 
meet   O'B  produced  ;    draw   also      „ 
00  parallel    and   equal  to  FG  ; 
join    0  and   0'.     From  triangle 
00' C, 

tan  y  =  — ,      ....    (120) 

2B 


•~-'c 


Fig.  60. 


00'  = =  2R  sec  y  =  \^^R''  +  ^'.   . 

cos  y 


(121) 


Then  in  triangle  00' M, 

O'M       2R  -  p 


cos  2   = 


y     \      ^Rj 


cosy. 


(122) 


00'       2R  sec 

Now  knowing  y  and  z, 

a  =  z-y (123) 

Next,  ic  =  oif=  00' sing  =  2i? secy  sin  z.      .     (124) 

Example.— Given  D=T  30',  ^  =  62  ft.,  I  =  100  ft.,  to  locate 
crossover  when  A  is  at  sta.  86  +  20. 

By  (120), 
log  tan  2^  =  2  -  3.18441  =  8.81559  =  log  tan  3°  44'. 

By  (121), 

log  00'  =  3.18441  -  9.99908  =  3.18533  =  log  1532. 

By  (122), 
log  cos  z  =  3.16643  -  3.18533  =  9.98110  =  log  cos  16°  47'. 

By  (123), 

a  =  16°  47'  -  3°  44'  =  13°  3'. 

By  (124), 

log  X  =  3.18533  +  9.46053  =  2.64586  =  log  442.4. 


106     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

133.  To  Find  the  Radius  of  the  Reversed  Curve  AFE,  Fig 

^O'  61,  Given  Angles   /  and  1' ,  and 

BG=k. 

From  the  figure, 

E  tan  ^I  =  BF, 
i?tany=  OF. 
Adding, 
•         i2(tan  ^1  +  tan  i/')  =  BG=k. 


Fig.  61. 


Whence 


R  = 


k 


(125) 


tan  U  +  tan  \I' 

Example.— Given  /  =  10°,  7'=  20%  BC  =  700  feet,  to  find  B. 

700 


By  (125),     B  = 


0.08749  +  0.17633 


=  2653  ft.,  a  2°  9f  curve. 


134.  To  Locate  a  Reversed  Curve  between  Fixed  Points. 
In  Fig.  62  let  AB  =  k,  and  angles  I  and  /   be  known.      We 


Lave  to  find  B  and  the  angles  a  and  b. 

Draw  O'G  parallel  to,  and  OG  and  O'F  perpendicular  to,  AB. 
Angle  AOG  =  I  and  BaF  =  I'.  Then  OF  =  B  cos  /  and  OF 
=  B  cos  /'.     Hence 

OG  =  i?(cos  I  +  cos  7'). 

In  triangle  00' G,  00  =  2B.     Therefore 

i?(cos  1  -r  cos  1')      cos  I  -f-  cos  I' 


cos  X  = 


2B  ~  2 

an  expression  from  which  B  has  disappeared. 


.     (12ff) 


LOCATION. 


107 


We  uow  have  a  =  I  -\-  x    aud     b  =  I'  -\-  x. 

To  tiud  i^  we  Lave  AE  -\-  EF  -\-  FB  =  k, 
or  i^  siu  /  +  2R  sin  x  -\-  R  sin  I'  =  k. 

Whence  R  — 


k 


.  ^    ^     ,     (127) 

sin  i -f- sin  7' -f  2  sin  05 '  * 

Another  expression  for  R  can  be  found  by  drawing  ^iVand  BL 

perpendicular   to    00',  and  BN  parallel   thereto.     Then,    since 

4  BAN=x, 

RsXn  a  -\-  R  sin  h  =i  k  cos  x. 

k  cos  X 


sin  a  -\-  sui  0 
Example. — Take  the  example  of  the  last  problem, 

A  =  700,     1=10°,    J'  =  20\ 
By  (126), 

cos  X  =  1(0.98481  4-  0.93969)  =  0.96225  =  cos  15°  48'. 

We  now  have        a  =  25°  48'     aud    b  =  35°  48'. 

700  X  0.96225 


(128) 


By  (128),     R 


=  660.2  ft.,  an  8°  41'  curve. 


0.4^523  +  0.58496 

135.  To  Connect  Two  Divergent  Tangents  by  a  Reversed 
Ourve. 

First  Case. — Advancing  towards  the  P. I. 

Given  the  radii  R  and  A*i  ,  the  angle  /  and  AG  =  k,Xo  find  the 
angles  a  and  b  (Fig.  63). 


^--^ 


A- 

^ 

***** 

Oy 

/ 

/ 

^^A^ 

^ 

z 

< 

R/i"^. 

>^ 

\ 

M"^ 

H 

^*'  / 

y 

"^ 

\,.^ 

mT^--^ 

Ey 

^^ 

okiF.. 

..t.  ' 

Q 

Fia.  63. 
Draw  00  parallel  to  the  tangent  BC  to  meet  O^B  produced. 
Then  EF  =  BO  =  AF -  AE. 

Therefore  BG  =  R  cos  I  —  k  sin  /. 


108     A   FIELD-MAIsrUAL   FOR   RAILROAD    ENGINEERS. 

From  triangle  00 iG, 

Ri  +  BG      Hi  4- RcosI- ksml  ,,^^, 

''''  =  -E+Mr'= BTB. ■•      •     ^''^^ 

Then  a  =  M0i2^  =  b  -  I,  O.Jf  being  parallel  to  OA. 

Second  Case. — Receding  from  the  P.I. 

In  Fig.  63  we  have  BC  =  ki  ,  angle  /,  R,  and  2?,  given,  to  find 
angles  a  and  b. 

Pioduce  OA  to  meet  OiL  drawn  parallel  to  CA.  AL  equals 
OiM=  O^H  cos  I. 

0,H  =  R,  -  EB  =  Ri  -  k,  tan  I. 

.*.    AL  =  OiM  —  {Ri  —  ki  tan  I)  cos  /. 
Hence 

OL  =  R  -{-  (Ri  —  kx  tan  7)  cos  /  =  i?  +  Rx  cos  I  —  k^  sin  7. 

From  triangle  OOiL, 


cos  a 


OL  Z?  +  7?i  cos  7  —  kx  sin  7 


00,  ~  i?  +  7?i 

Evidently,  5  =  a  +  7. 


.  .     .    (130) 


136.  To  Change  the  P.R.C.  so  that  Second  Branch  of 
Curve  shall  End  in  a  Tangent  Parallel  to  Terminal  Tangent 
and  Distant  p  therefrom. 

In  Fig.  64  let  MAB  be  the  located  curve,  EN  =  p.    We  must 


M 


H 


E    F 


.-a- 


/ 

1     ^ 

V  B                  !n  JQ 

— > 

._. 

NO 

lL ^- 


determine  the  angle  CO  A,  after  which  the  desired  curve  AGE 
may  be  located. 

Draw    HOx  and  LOx  parallel  to  ^Fand  NG. 


HL  =  0,K  =  p. 


LOCATION. 


109 


From  triangles  00, '77  and  OO^L, 

{R  +  Ri)  cos  b  =  (R-\-  Rx)  cos  a  -  p. 

P 


. '.  cos  5  =  cos  a 


R-\-Rx' 


(131) 


Angle  AOG  =  b  —  a. 


137.  To  Find  the  Radius  of  a  Curved  Track. 

Measure    any  chord    AB  =  21,   and  mid-ordinate   CE  =  M. 


Fig.  65. 
Theu  in  the  right  triangle  OAE  (Fig.  65), 

72^-  (7?  -  Mf  =  ?. 


.  R 


2M 


(132) 


\ 


CHAPTER  IV 


TRANSITION-CUR  VES. 


Article  11. — Theory  op  the  Transition-curve. 


138.  Elevation  of  Outer  Rail  on  Curves. — To  counteract  the 
effect  of  centrifugal  force  on  curves  the  outer  rail  must  be 
elevated  above  the  inner  one.  It  is  shown  in  mechanics  that  the 
centrifugal  force  is 


F  = 


32.16i?' 


where  W  is  the  weight,  v  the  velocity  in  feet  per  second,  32. 1(^ 

an  average  value  of  the  acceleration  of  gravity  in  feet  per  second 

per  second,  and  R  the  radius  in  feet. 
In  Fig.  66  let  the  vertical  BL  represent  W,  the  horizontal  KH 

the  centrifugal  force,  AB  the  plane  of  the  rails,    and  GB  =  e 

the     superelevation     of    outer     rail 
From  similar  triangles, 

Equate  this  value  of  F  to  that  given 
above  and  solve  for  e,  giving 

^^^'  .     (133) 


K,*F*iH 


e  = 


32.1622' 


The  gauge  AB  should  be  greater  on  curves  than  on  tangents 
to  allow  for  flange  clearance  and  the  effect  of  a  rigid  wheel-base. 
AG  =  4.9  feet  is  about  the  right  value  for  the  horizontal  distance 
between  centers  of  rail-heads  for  standard  gauge.  In  formula 
(133) ■«  is  in  feet  per  second,  but  the  train  velocity  is  usually  given 
in  miles  per  hour.     Let  V  =  velocity  in  miles  per  hour,  then  the 

110 


TRANSITION-CURVES.  Ill 

22 
velocity  in  feet  per  second  will  be  «  =  —  F.     Inserting   these 

values  in  (133)  gives 

4.9X484F*        V  .  ,,„.. 

This  elevation  will  be  required  from  the  P.C.  to  the  P.T.,  but 
obviously  it  cannot  be  introduced  suddenly,  so  that  for  easy 
riding  the  rate  of  increase  of  e  should  be  uniform.  From  (134)  it 
is  seen  that  e  varies  inversely  with  R,  which  requires  that  when 
e  =  0,  R  =  infinity.  Hence  II  must  decrease  from  infinity  to 
the  radius  of  the  circular  curve,  while  e  increases  fron\  0  to  its 
maximum  value. 

139.  The  True  Transition- curve  should  satisfy  formula  (134), 
but  so  far  no  such  curve  has  been  found  that  will  at  the  same 
time  admit  of  the  same  ease  of  location  as  the  simple  circular 
curve.  According  to  Rankine  the  first  use  of  any  other  than 
the  circular  curve  was  made  by  Gravatt  about  1828  or  1829, 
the  curve  employed  being  the  curve  of  sines.  Another  method 
described  by  Rankine  is  attributed  to  William  Froude  about 
1842  ;  this  curve  was  worked  up  in  the  Engineering  News  by 
A.  M.  Wellington  in  1890.  Other  approximations  are  the  Rail- 
road Spiral,  developed  by  W.  H.  Seailes  in  1882,  and  the  cubic 
parabola,  described  by  C.  D.  Jameson  and  E.  W,  Crellin  in  the 
Railroad  and  Engineering  Journal,  1889. 

In  1880  Ellis  Holbrook  described  in  the  Railroad  Gazette  the 
true  transition-curve  applicable  to  small  angles  and  short  lengths 
of  the  curve.  In  1893  C.  L.  Crandall  published  formulae  and 
tables  applicable  to  large  central  angles  for  both  the  offset  and 
deflection  methods. 

140.  The  Notation  here  employed  will  be  explained  with 
reference  to  Fig.  67.  The  curve  CBB'C  is  the  circular  curve 
offset  at  Cand  C"  from  the  tangents  by  the  amounts  (7// and  C'll'. 
AGB  and  B'G'A'  are  the  transition-curves.  A  is  the  P. 7'. C, 
or  point  of  transition-curve,  C  the  P.O.,  B  the  P.O.,,  B'  the 
P.TC.i,  G  the  P.T.,  aud  A'  the  P.  7'.,.  The  co-ordinates  of  G 
are  All  =  x',  HG  =  y';  of  C,  x'  and  TIG  =  F ;  of  B,  AM  =  Xi 
and  MB  =  y^.  The  length  of  curve  from  P.  T.  C.  to  any  point  P 
is  I,  aud  the  whole  length  from  P.T.C.  to  P  d  is  ^i. 


112     A   FIELD-MANUAL   FOR   KAILKOAD    EXGINEERS. 


141.   Equation     of    Transition- curve. — Since     the     rate     of 
change  of  e  must  be  uniform,  (134)  may  be  written  • 


e  =  kl  — 


3p' 


(135) 


Fig.  67. 


in  which  k  is  the  rate  of  rise  of  outer  rail  along  curve,  and  p  the 
varying  radius  of  curvature.  From  the  calculus  pd(p  =  dl, 
whence 


dl 


(136) 


Insert  this  in  (135)  and  solve  for  d(p. 

3yfc 
d(p  =  -y^/dl  =  2mldl, 


(137) 


2m  is  dependent  upon   V  and  k,  and  is  constant  for  any  one 
curve. 

Integrating  (137), 

<p  =  ml\ (138) 

the  constant  of  integration  being  zero,  for  I  is  zero  when  cp  is 
zero. 


TRANSITION-CURVES.  113 

From  the  elemenlary  triangle  drawn  at  the  point  F  of  transi- 
tion-curve, Fig.  67,  dl  being  tangent, 

^y       •    ^ 

~  =  sm  0. 
al 

Expanding  sin  (p  by  trigonometry, 

in  which  3!  =  3x2x1,  5!  =  5x4x3x2x1,  etc. 
Substituting  for  0  its  value  from  (138), 

dy  =  dl\ml^  _  -^  +  _—  _  ^^  +  .  .  . j. 
integrating, 

^=  ^X"  "42- +  1320  -75600+ •••/    *     *     ^^^^' 

Jmt  mP  =  <p,  where  0  is  in  circular  measure.     To  obtain  0  in 
cf«gre6s,  *p  ^  9*°T^  =  ^^^-5-     Inserting  this  in  (139), 

^OU         oi.O 

^         Vl'71-89      "79  X  10*  "*"  8151  X  10«       153245  X  10»^  -^  "  'J' 

or  y  ^IG, (140) 

in  which  G  may  be  found  from  "fable  XIV  with  0°  as  argument. 
Interpolation  must  be  resorted  to  for  values  of  0  not  given  in  the 
table,  or  y  computed  by  the  formula. 

From  the  elementary  triangle  at  P,  Fig.  67 

dx 

-—  =  cos  0. 

al 

Expanding  by  trigonometry. 


114     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 
Substitiitiug  ml'^  for  0  and  integratiug, 

"^-V       1^"^  216"  ""9360  +••  -y ^^^^^ 

Replacing  mP  by  (p  reduced  to  degrees, 

~    \         32828  ^  2328  X  10«      33114  X  10'°  ^  ■  '  'j 

or  X  =  I  —  IE. (142) 

Ovaries  with  (f)° ,  and  may  be  taken  from  Table  XIV  with  0° 
as  argument. 

142.  The  Transition-curve  Angle  /j  is  the  value  0  assumes 
attheP.Ci.     From  (138), 

Ii  =  mh^ (143) 

From  (137)  and  (136), 

dl^ 1_  _ 

d(p  ~  2ml  ~  ^' 

5730 
At  the  P.C.i  p  =  R  and  may  be  taken  equal  to  -jr^,  so  that 

1  „      5730 


2mli       ^'        D"  ' 
whence 

1  D" 


""' =  2i;r=  lurn. (1^) 


This  value  of  m  in  (143)  gives 

272^11460 

Reducing  this  to  circular  measure  by  writing  Ii=Ii 


o     ^  U 


180~57.30 
gives 

ir  -  28.65^  =  1°-' (146) 

143.  The  Coordinates  of  any  point  on  the  curve  are  given  by 
(140)  and  (142).     The  length  of  the  transition-curve  being  known 


TRANSITION-CURVES.  115 

or  assumed,  y^  and  Xi  (the  coordinates  of  the  P.Ci)  may  be 
found  from  these  equations  by  the  help  of  Table  XIV;  the 
coordinates  of  the  P.  C.  (see  Fig.  67)  will  be 

F=y,  -  7?(1  -  cos  /i)  =  y,-R  vers  /, ,   .     .     (147) 
x'  =  Xi-R^mIi (148) 

144.  Deflection-angles. — With  the  transit   at  the  P.  7^.  C.  (or 
P.  T.  1  in  backing  up)  the   tangent  of  deflection-angles  may  be 

found  from  the  relation  tan  d  =  -.     Dividing  (139)  by  (141), 

tan  5  =  -—  +  .009523w3^«  +  .000167w5^i»  + (149) 

o 

From  trigonometry  the  expansion  of  the  angle  in  terms  of  its 
tangent  is 

8  =  tan  <^  —  ^  tan=*  ^  ■{-  \  tan*  8  —  etc.      .     .     .   (a) 

In  (149)  write  mV^  =  (p  and  substitute  in  (a) : 

5  =  -|  -  .0028230'  -  .00006805 (150) 


From  (138)  and  (143), 


in  which  --  =  n.    From  (5),  0  =  IiV?,  and  this  in  (150)  gives 

Ox 

d  =  ^n'^  -  .002823/i37i«  -  M00Q8L'n">.  .     .    .     (c) 

o 

Both  S  and  7i  are  in   circular  measure  ;  to  reduce  to  degrees 

TT 

multiply  by  -^.     This  gives,  neglecting  terms  involving  higher 
powers  ot  7i  than  the  third, 

5"  =  :^  71-^-  .00000086  7i 3,16 (151) 

The  second  term  is  quite  small,  and  in  most  cases  may  be  en- 
tirely neglected  in  practice. 


116     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

With  the  instrumeut  at  auy  intermediate  point  x"y"  the  deflec- 
tion-angle for  any  point  xy^  measured  from  initial  tangent,  will  be 

tan  8  =  ^-^,  =  i(mP  +  ml"'  -f  mil")  +jh?{mH'  +  mH"') 

-f  ^%{mHH"  4-  mHl"^)-{-  ^^^^{mHH''^  +  mHn"^+mHH"^)+  ...,  (152) 

in  which  powers  of  mP  higher  than  the  third  have  been  neglected. 
Substitute  the  value  of  tan  d  from  (152)  in  (a),  write  ml^  =  0  = 
7,w^  ml'"^  =  <p"  =.i,7i'"^  by  (6),  and  reduce  circular  measure  to 
degrees,  giving 

I " 
d^  =  -^-(w'  +  ^"'  +  ^^")  —  a  small  correction.        (153) 


For  instrument  at  P.T.C.,  n"  =  0  ;  then  (153)  yields 

(^o")  =  -^7i'  —  correction, 
o 

or 

{§:)  =  ^A,  -B, (154) 

(154)  is  the  same  as  (151),  as  it  should  be. 
For    the    transit    at     the     quarter-point    of    tr£insition-curve 

n"  =  Y  =  ip  =1;  then  (153)  yields 

ll  il  4 

(^i")  =  Y^'''  +  ^'«  "^  ^""^  ~  correction. 


or 


(V)  =  ^°^j  -  5j (i5r>) 


For  transit  at  mid-point  of  transition-curve  n"  =  |,  and,  from 
(153), 

7  ° 
(^i°)  =  -5-(^'  4-  i  +  i^)  -  correction, 


or 


(5^°)  =  ^-^A^  -B^ (156) 


TRANSITION-CURVES. 


117 


For  transit  at  three-quarter  poiut  n"  =  |  and 


or 


(^D  =  -t(^*'  +  T5  +  f^)  -  correction. 


(5j°)  =  —^A^  —  B^ •••    (157) 


For  transit  at  P.C.i  7i"  =  1  and 


// 


or 


(5x°)  =  -^(w'  +  1  +  w)  -  correction, 
o 


/i' 


(5.-)  =  :^^, -5,. 


(158) 


With  the  transit  at  the  P.T.d  it  will  frequently  be  most  con- 
venient to  measure  the  deflections  from  the  tangent  to  the  circular 
curve  at  that  point.  Sometimes  this  will  also  be  the  case  for  the 
transit  at  the  P.C.i. 

Bv  reference  to  Fig.  68  it  will  be  seen  that  for  the  transit  at  B 


the  deflection  from  the  tangent  BC  which  serves  to  fix  any  point 
on  the  curve,  as  .6,  is  given  by  the  equation 

or,  in  general, 


(<5c°)  =  —Ac  +  B,. 


(159) 


118     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

Table  XV  gives  the  values  of  A  and  B  for  the  five  positions  of 
instrument  for  which  equations  (154)  to  (159),  inclusive,  were  de. 

duced.     The  value  of  A  must  be  multiplied  by  -^ ,  but  B  is  taken 

o 

direct  from  the  table  in  thousandths  of  a  degree. 

If  deflection-angles  are  wanted  for  other  positions  of  the  instru- 
ment, or  for  other  points  on  the  curve,  they  may  be  computed 
from  equation  (153). 

145.  Tables. — Three  tables  are  given  for  use  with  transition- 
curves. 

Table  XIV  was  computed  for  use  with  formulas  (140)  and  (142) 
in  determining  C  and  E ;  0  being  assumed  and  G  and  E  com- 
puted. .  ^ 

Table  XV  gives  A  and  B  for  computing  the  deflection-angles 
by  (154),  (155),  (156),  (157),  (158),  and  (159)  for  20  equidistant 
stations  on  the  transition-curve.  For  points  not  given  in  the 
table  A  and  B  must  be  interpolated.  Linear  interpolation  will 
suffice  in  most  cases,  though  when  Ii°  is  quite  large  second  differ- 
ences may  be  preferable  for  A.  B  is  given  in  the  table  in  thou- 
sandths of  a  degree. 

Table  XVI  was  calculated  by  assuming  h  in  lengths  varying 
by  increments  of  20  feet,  then  computing  7/  by  (146),  yi  by  (139), 
X,  by  (141),  F  by  (147),  and  x'  by  (148).  y,  and  x,  will  also  be 
given  more  directly  by  (140)  and  (142)  with  the  aid  of  Table  XIV. 

The  excess  in  length  of  transition  curve,  measured  from  P.T.C. 
to  the  point  on  offset  at  B.C.,  over  x'  is  tabulated  as  e;  I'  is 
found  by  trial  such  that  when  inserted  in  (141)  or  (142)  the  same 
value  of  x'  will  be  obtained  as  in  (148).     This  may  be  done  by  as- 

suming  l'  a  little  less  than  -  ,  then  computing  .i'.     More  than  two 

trials  will  rarely  be  needed  to  find  a  sufficiently  close  value  of  l'; 
then  e  =  I'  —  x'.  y'  is  found  by  (139)  after  finding  l',  or  0' 
may  be  found  from  (b)  of  144,  and  used  in  (140)  in  connection 
with  Table  XIV.  h  -  I'  is  the  length  from  G  (Fig.  67)  to  the 
P.C.i;  the  difference  in  length  between  this  and  the  length  of 
circular  curve  from  P.  C.  to  P.  G.  i  is  tabulated  as  e' ;  that  is,  e'  = 
{li  —  V)  —  arc.     Then  e -\-  e  =  U  —  {x'  -\-  circular  arc). 

For  values  of  li  intermediate  between  those  given  in  the  table 
linear  interpolation  will  suffice,  1  hough  second  differences  may 
be  used  for  ii^and  yi  if  ])ref erred. 


TRANSITION  -CU  RYES. 


119 


146.  To  Unite  the  Two  Branches  of  a  Compound  Curve  by 
i\  Transition-curve. 

The  same  objections  bold  to  compound  curves  as  to  simple 
curves  uniting  with  a  tangent;  i.e.,  wbere  there  is  a  sudden 
change  of  curvature  there  should  be  a  sudden  cbange  of  super- 
elevation of  outer  rail,  which  of  course  is  not  allowable.  Instead 
of  compounding  the  curves,  we  may  offset  them  at  theP.  C.  C. 
and  unite  them  by  means  of  a  portion  of  a  transition-curve  tangent 
to  each  of  the  simple  curves. 

In  Fig.  69  AB  am]  CELMare  the  simple  curves  that  are  to  be 
united  by  the  transition-curve  ANE,    Extend  the  transition-curve 


Fig.  69. 

to  O,  where  its  radius  of  curvature  becomes  infinite,  and  let  G8 
be  its  tangent.  Call  the  length  of  transition-curve  from  G  to  A 
li  ,  from  O  to  E  h ,  and  from  E  to  A  h.  E  and  A  are  points 
of  tangency  of  simple  and  transition  curves.  Then  l^  =  It  —  U. 
The  coordinates  of  A  arc  08=  Xi ,  SA  =  y^  ;  and  of  V{WV 
perpendicular  to  GS),  GW=x,',  WV  =  F,;  of  E,  GP  =  x, , 
EP  =  y^',  of  L  {LH  perpendicular  to  GS),  GH  =  Xs,  EL  =  Fz. 
Let  BC=F^. 

The  radius  of  curvature  of  transition-curve  is  inversely  pro- 
portional to  its  length  from  G  ;  hence  the  curvature  is  propor- 
tional to  the  length  of  curve;  therefore  ^3  :  h  =  B3  :  Di  ,  whence 


^3  =  I, 


B,' 


(160) 


120     A   FIELD-MAXUAL   FOR   RAILROAD    ENGINEERS. 
Then         h -.    U  -  u  =  IM  -  -~\  =  I,    '  ~^    \   .     .     (161) 

By  (138)  or  (143),  A  =  U  )  * 

Equaling  the  value  of  /a  from  this  equation  to  that  resulting 
from  (146)  gives 

^'  =  ^^U)  =  -200 (162) 

WV  =  Fi  and  RL  =  F^  may  be  taken  from  Table  XVI  with 
h  and  ^3  as  arguments.  Then  0,  W  =  R,  +  F,  O^H—  R^-^-Fz. 
Draw  0,r parallel  to  QS,  then  0,T  =  WH;  henc6 

0,T  =  (R,  +  F,)  -  (R,-{.F,), 

and  OlT=Xl'-a^'. 

Therefore 

'^^-  =  iR,  +  F.)-iR.+F.)'      .    .     .    (163) 

0.0,  =  {x,'  -  a-s')  cosec  a  =  VoJ''  +  T07\      .     (164) 
Then  CB  =  CO,  -  BO,, 

or  i^2  =  i?s  -  (i?i  + OiO,) (165) 

The  lengths  of  AB  and  CE  are 

^^  =  ^'°^^''°100 (166) 


[3" 


CE=  ^-^^m (167) 

The  excess  of  transition-curve  length  over  AB  +  C^is 

e,  =  h-~i—^^ 1 B, )  •     •     •     (16^) 


TRANSITION-CURVES. 


121 


If  AB  and  CE  are  quite  sharp,  we  must  take  account  of  the 
arc  excess,  so  that  we  have  then 


Cj  —  frj  — 


f^'°^   ^  +  ^^-^  J  100  +  arc  excess    | .     (168') 


The  arc  excess  may  be  taken  from  the  second  column  of 
Tuble  IV,  which  gives  the  arc  length  for  one  station;  this  multi- 
plied by  the  number  of  stations  gives  the  curve  length,  which 
may  replace  the  values  within  the  brackets  in  (168'). 

147.  Length  of  Transition-curve  to  be  Taken. — In  practice 
the  rate  of  change  of  superelevation  of  outer  rail  may  vary  from 

VHhV  ^^  400*    ^^^^  ^^^  ^^^^  ^  '  ^^^^  evidently  kl,  must  equal  the 
superelevation  of  outer  rail  for  circular  curve  ;  or,  by  (135), 

72 


Writing  i?  = 


5730 
I) 


and  solving  for  li 


U 


For  k  = 
For  k  = 


1200' 

1 

600' 


17190A; 


(169) 
(169') 


^^'  ^  =  400' 


li  =  0.035  F*i> (169") 


h  =  0.02s  V^D (169'") 


The  following  table  gives  values   of  Z,  in  feet  per  degree  of 
circular  curve  for  a  few  values  of  Fand  k. 


k 

30  Miles 
per  Hour. 

35  Miles 
per  Hour 

40  Miles 
per  Hour. 

45  Miles 
per  Hour. 

50  Miles 
per  Hour. 

55  Miles 
per  Hour. 

1 
1200 

C3 

86 

112 

142 

175 

212 

1 

600 

32 

43 

56 

71 

83 

106 

1 

400 

21 

29 

37 

47 

58 

71 

122     A    FIELD-MANUAL    FOR   BAILROAD    ENCiLNEERS. 

"\Ybeu  only  a  short  tangent  intervenes  between  two  curves 
shorter  transition-curves*  must  be  takeu,  requiring  larger  values 
of  A*,  so  that  overlapping  may  be  prevented. 

For  illustration  suppose  a  5"  curve  to  be  eased  off  with  a  tran- 
sition-curve, the  highest  train-speed  being  45  miles  per  hour  and 

k  =  --.     By  the  table  the  value  of  h  will  be  71  X  5  =  355  feet, 
dOO 

so  that  we  should  probably  take  a  360-ft.  transition -curve,  re- 
quiring an  offset  of  4.7  feet  by  Table  XVI. 

Article  12. — Field-work. 

A      Field  Formulas. 

148.  For  the  cases  most  frequently  presenting  themselves  in 
practice  the  foregoing  formulas  may  be  simplified  so  as  to  admit 
of  the  rapid  location  of  points  on  the  transition-curve  with  all  the 
accuracy  needed  on  location,  though  it  is  best  to  use  the  exact 
formulas  and  tables  in  setting  track-centers  on  the  finished  road- 
bed. "When  the  transition-curve  aogle  is  quite  large  it  will  be 
better  to  use  the  accurate  methods  on  location  also,  but  for  the 
more  common  cases  the  followiuc:  formulas  will  answer. 


■o 


149.    Simplified    Formulas.— In   (189)  and   (140)   neglect,    as 
small,  all  the  terms  following  the  first,  giving 

y  =  '^  =  ^l=  Mh^m^" (170) 

In  (141)  and  (142)  retain  only  the  first  two  terms 

^=^^"llf)   =  ^(l  -  ^)  =  ^ -- -OOOOSZ^^S    .     (171) 

in  which  the  last  term  is  small  for  short  transition-curves  and 
may  often  be  neglected,  x  being  taken  equal  to  I. 
The  values  of  m  and  /i  remain  as  before  : 

2Rli       11460^1 

^■"=2««4=l^ ("«^ 


TRANSITION-CURVES.  123 

In  (147)  expand  cos  7, ,  giving 


F-y.-Ii{^-^  "  24  +  720"'--/' 

But  7?  =  -^,  by  (145).     Substitute  this  for  E  and  neglect  all 
but  the  first  two  terms  : 


But  i,/i  =  3y, ,  by  (170),  since  0  =  /i  and  j^  =  yi  when  Z  =  ^i  ; 

hence 

F=yx-^y,  =  iyt (173) 

Likewise  expanding  sin  7i  in  (148), 

nnd  writing  R  =  -~  as  above, 

« 
But  x^=l,-  -^- ,  by  (171). 


By.  (170), 


^  -  3V2J         3  "^   8         8        2 


In  (154),  (155),  (156),  (157),  (158),  and  (159)  neglect  the  correc 
tion;  then 

(5o°)  =  ^^o (175) 

(.V)-^"^i (176) 


]24     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

(V)  =  ^^i.        ......     (177) 

(V)  =  ^°^i (178) 

{^,^)  =  ^A^ (179) 

iSc'')  =  ^Ac (180) 

160.  Offsets. — Formula  (170)  shows  that  offsets  from  transition- 
curve  to  tangent  vary  as  the  cube  of  the  distance  from  the  P.  T.C., 
and  it  can  be  shown  that  offsets  from  the  circular  curve  to  tran- 
sition-curve follow  the  same  law,  reckoning  from  the  P.G.i. 

Formula  (36)  may  be  written 

z=f{l'D), (a> 

in  which  D  is  the  degree  of  curve  if  offset  is  from  tangent,  and  the 
difference  of  degrees  if  offset  is  ])etween  two  curves  having  a  com- 
mon  point  of  tangeucy,  I  being  reckoned  from  the  tangent-point. 
From  (136)  and  (137), 

^  -      _  J_ 

and  the  degree  of  transition-curve  at  any  point  is 

j)^  =  "il^  =  lUQOml  =  d (181) 

P 

Formula  (181)  shows  that  the  degree  of  curvature  of  transition- 
curve  at  any  point  is  a  function  of  its  length.  If  the  D  in  (a)  is 
the  difference  between  degrees  of  circular  and  transition  curves,  it 
will  equal  Di  —  Dt ,  which  is  also  a  function  of  the  length;  so 
in  {a)  write  D  =f{l),  giving 

z=f'{l% (182) 

which  shows  that  the  offset  betw^een  circular  and  transition  curves 
varies  as  the  cube  of  the  distance  from  P.d.  The  offset  at  the 
P.  0.  is  known,  being  half  of  F,  and  may  therefore  be  found  for 


TRANSITION-CURVES.  125 

Other  points  ;  thus  midway  between  P.C.  and  P.C.i  it  will  be  one 
eighth  of  its  value  at  P.C,  or  ^^F 

151.  Compound  Curves. — By  trial  it  has  been  found  that  <?, 
^see  formulas  (168)  and  (168')— equals  e  +  e'  from  Table  XVI 
when  the  table  is  entered  with  D  —  Bx—Dz  and  U  as  arguments, 
up  to  about  BiU  =  8000,  which  covers  all  cases  in  ordinary  rail- 
road practice  ;  so  we  write 

e-t=e  +  e' (183) 

The  distance  AB  on  sharper  curve  is  found  by  trial  to  equal 
^^a  up  to  about  Bill  —  4000,  which  answers  for  the  ordinary  cases 
arising  in  practice.     When  Bili  is  greater  than  this  AB  (of  Fig 
69)  must  be  found  from  (166). 

The  point  JV  can  be  taken  midway  between  G  and  B,  for  the 
radius  of  transition-curve  decreases  uniformly  from  Ea  to  Mi  and 
may  be  here  taken  as  their  mean  ;  hence  the  offset  BN  =  NC  = 
^F-2.  Other  offsets  may  be  found  from  the  relation  given  by  (182). 
Thus  the  offset  midway  between  A  and  B  will  be 


^        2   \il4        16 
Other  offsets  may  be  obtained  if  desired 


3 

16 


B.  Setting  Out  Transition -curves. 

152.  In  first  locating  the  line  it  will  be  sufficient  to  simply 
offset  the  curve  at  the  P.  0.  the  amount  required  for  the  transition- 
curve  ;  then,  with  the  transit  over  this  point,  bring  the  telescope 
parallel  to  the  tangent  from  which  offset  was  taken  and  run  the 
circular  curve  to  the  P.2\,  where  another  offset  is  made  and  a 
tangent  parallel  to  the  terminal  tangent  of  circular  curve  can  be 
run  out.  The  amount  to  offset  will  be  governed  by  the  length  of 
transition-curve,  or  if  the  offset  is  fixed  it  governs  the  length  of 
curve.  Either  li  or  ii^  being  given,  the  other  may  be  taken  from 
Table  XVI. 

153.  Location  by  Oflfsets. — If  the  offset  is  given,  ^i  can  be  taken 
from  Table  XVI,  interpolating  if  necessary.  At  the  P.  C.  bisect 
the  offset,  and  set  a  stake  at  that  point ;  then  measure  back  along 
tangent  the  distance  x',  which  for  most  cases  may  be  taken  as  ^^i 


126     A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

(see  formula  (173) ),  and  set  a  stake,  marking  it  P.T.C.  From  the 
P.  C.  measure  forward  around  the  circular  curve  a  distance  equal  to 

/  ° 

~,  which  approximately  equals  Ui.     Set  a  stake  marked  P. (7. . . 

At  the  quarter-point  offset  from  tangent  an  amount  equal  to  j^F, 
for,  by  (170),  the  offsets  are  proportional  to  the  cube  of  the  dis- 
tance from  P.T.C. ,  so  that 

^       2{lh)^      IC   • 

At  the  three-quarter  point  offset  the  same  amount  from  circular 
curve.  If  the  transition-curve  is  not  over  400  feet  long,  these  are 
all  the  points  need  ;  if  longer,  other  offsets  are  similarly  found 

Example.— At  sta.  412  an  offset  of  4.2  feet  was  made  from  a 
tangent  to  a  5°  curve.  Required  the  data  for  a  transition-curve 
to  connect  tangent  and  circular  curve. 

By  Table  XVI  it  is  seen  that  a  340-ft.  transition-curve  is  re- 
quired. From  the  table  it  is  seen  that  x'  =  169.9  ft.,  Ii°  =  8.5°, 
and  excess  of  curve  over  tangent  is  .02  ft.,  which  we  neglect  as- 
small.  Drive  a  stake  2.1  ft.  from  offset  hub  and  mark  it  412  ; 
measure  back  along  tangent  169.9  ft.  to  410  -\-  30.1,  and  drive  a 
stake  marked  P.T.C.     Measure  forward  around  circular  curve 

8  5 

-4-  =  1-70  chains  =  170  ft.,  and  set  a  stake  marked  P.C.i  at  sta. 
5 

413  +  70.1 

The  approximate  offsets  are  : 

4.2 
At  mid-point,  sta.   412,  ^  ~  ~o'  =2.1 

*'  one-eighth  points, stas.  4{3  it  27  g  [  .     t  =  2.1  X(i)'=  0.033 
"  quarter-points,      stas.  ^J^  +  gsj  \>     ^  =  0.033x23=  0.26 


"  three-eighths  points,  stas.        i  40  fii'  <  =  0.033  x  3'  =  0. 


89 


Stakes  at  the  one-eighth  and  three-eighths  points  were  not 
needed,  but  were  worked  out  for  illustration. 

154.  Location  by  Deflections. — The  number  of  chord-lengths 
being  taken  as  an  aliquot  part  of  20,  the  deflection  angles  for  the 


TRANSITION-CURVES.  127 

transit  at  any  one  of  five  positions  may  be  taken  from  Table  XV  by 

1  ° 
inultiplyiug  the  tabular  values  of  A  by  -^,  Ji"  being  found  from 

o 

Table  XVI  or  formula  (146).  If  the  number  of  chords  is  not  an 
aliquot  part  of  20,  or  if  the  transit  is  at  some  point  other  than  one 
of  the  five  for  which  Table  XV  was  calculated,  then  the  deflec- 
tion-angles must  be  computed  by  (153).  The  curve  is  then  run 
out  in  the  usual  way. 

When  7i  is  not  more  than  15  or  20  degrees  the  curve  may  be 
run  from  the  P.T.G.  or  P.T.G.i  by  neglecting  the  correction  B  as 
small.  Even  when  7,  is  greater  than  20°  the  correction  may  be 
neglected,  provided  half  the  transition-curve  is  run  from  the 
P.T.G.  and  the  remainder  with  the  transit  at  the  mid-point,  the 
telescope  being  first  placed  parallel  to  original  tangent. 

Example. — Take  the  example  of  the  last  section:  li  =  340  ft., 

340  X  5 
F  =  4.2  ft.,  D  =  5°.    By  formula  (146),  L  =      ^^^  '   =  8.5%  the 

L° 
same  as  given  by  Table  XVI.    Then  --  =  2.833°.    Divide  h  into 

o 

5  parts  of  68  ft.  each,  which  will  be  the  chord-length  to  be  used. 
From  Table  XV  for  transit  at  P.  T.  G.  the  deflections  will  be  : 

For  sta.  410  -f  30.1,  P.T.G.,  {d,°)o  =  0. 

"      "    410  +  98.1,  (5/).2  =  2.833  X  .04  =  0.1133  =  0°    6.8'. 
"      "    411  +  66.1,  (5/).4  =  2.833  X  .16  =  0.4533  =  0°  27.2'. 
"      "    412  -\-  34.1,  (<5o°).6  =  2.833  X  .36  =  1°    1.2'. 
"      "    413  +  02.1,  (5o°). 8  =  2.833  X  .64  =  1°  48.8' 
"      "    413  +  70.1,  ((5„°).    =  2.833  X  1     =2°  50'. 

Having  set  out  the  transition-curve,  move  to  P.  G.i  at  sta.  413  + 
70.1,  backsight  to  P.T.G.i ,  and  deflect  7,°  -  (V)i  =  8°  30'  - 
2°  50  =  5°  40',  and  run  out  the  circular  curve  to  the  P.  7.(7. i, 
which  suppose  to  fall  at  sta.  420.  Set  the  transit  at  this  point,  and 
cause  the  vernier  to  read  zero  when  the  telescope  is  in  tangent  to 
circular  curve.    The  deflections  taken  from  Table  XV  will  now  be : 

For  sta.  420  +  68,  (5o°).8  =  2.833  X  .56  =  1°  35.2'. 
"  "  421  -f  36.  (V).6  =  2.833  X  1.04  =  2°  56.8'. 
"  "  422  +  04,  (5e°).4  =  2.833  X  1.44  =  4°  4.8'. 
**  "  422  -f  72,  (5c°).2  =  2.833  X  1.76  =  4°  59.2'. 
«♦      "  423  4_  40,  (V)o   =  2.833  X  2       =5°  40'. 


128     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


Set  transit  at  423  +  40,  the  P.T.i,  backsight  to  420  and  deflect 
8°  30'  —  5°  40'  =  2°  50',  when  the  telescope  will  be  in  tangent. 

155.  Form  of  Transit  Notes.— The  following  will  illustrate 
a  form  of  notes  that  will  be  found  to  answer. 

Let  the  P.C.  of  a  4°  aurve  be  at  sta.  160  -f  50,  and  a  200-ft. 
transition-curve  be  employed.  Let  the  intersection-angle  I  be 
20°.  By  Table  XVI,  F  =  1.16  ft.,  7,°  =  4°,  x  =  100  ft.,  so  that 
P.T.G.  is  at  159  +  50.  Take  four  50-ft.  stations  on  transition- 
curve  and  determine  the  deflection-angles  as  in  the  last  section. 


Sta. 

Deflec- 
tion- 
angle. 

Central 
angle. 

Calcu- 
lated 
Course. 

Mag- 
netic 
Course. 

Remarks. 

167 
-1-50  0 
166 

-\-m 

165 

F.r.i 

2°  40' 
2°  15' 
1°40' 
0°55' 

4''0' 

-hso© 

164 
16.3 
16> 

F.  T.C.I 

"6°'0'   ' 
5°0' 
3°0' 
1°0' 

12°  0' 

+50© 

161 
-fSO 

160 
-+-50© 

159 

158 

P.C, ,4  C.L. 
P.T.C. 

"]'°*20' 
OMS' 
0°  20' 
0»5' 
0°0' 

4°0' 

Set  ver.  at  2°  40'. 
B.S.  to  1.59  +  50,  and 
deflect  to  0°.  Run 
circular  curve. 

li  =  200.  F=  1.16. 

20  —  2  X  4 
The  length  of  circular  curve  was  j =  3  stations,  since 

the  central  angle  was  12°.  With  transit  at  161  -f  50  set  the 
vernier  to  2°  40 ,  backsight  to  159  -h  50,  and  deflect  into  tangent 
with  the  vernier  reading  zero.  With  the  transit  at  164  -\-  50 
cause  the  vernier  to  read  zero  when  the  transit  is  in  the  tangent  to 
circular  curve,  and  run  the  last  transition-curve  by  deflections 
from  this  tangent.  With  the  transit  at  166  +  50  backsight  to 
164  +  50  and  deflect  4'  -  2°  40'  =  V  20',  when  the  telescope  will 
be  in  tangent  and  the  line  may  be  continued. 


TRANSITION-CURVES. 


129 


Article  13.    Transition-curve  Problems. 

166.  To  Find  the  Tangent  Distance  and  External  when 
Transition-curves  are  Employed,  Offsets  Equal. 

In  Fig.  70  let  AB  be  the  circular  curve,  EF  and  OH  the  tran- 
sition-curves. Let  EK  —  IIK  =  Ti  be  tlie  tangent  distances,  and 
J^K=  El  the  external  required.     Let  LK  =  T' .     Draw  PF  per- 


V^—^-^x: 


pendicular  to  LK  \    then  in  triangle  PVK,    FJ?  =  JP' tan  \I\ 
LK  =  AP  -\-  VK  is  now  known,  or 

T'  =  T+FianiL (184) 

Hence 

Ti=x' +  T'  =  X' +  T+ Ftan^L  .     .     .     .     (185) 

In  triangle  PVK,    PK  =  Fsec  |/,  so  that,  letting  PiV^  =z  E, 

Ei  =  E+FseciI (18G) 

Example.— Two  tangents  intersect  at  sta.  91  +  37  8;  required 
the  tangent  and  external  when  F  —  2.62  ft.,  /  =  26°  30',  D  =  4". 

By  Table  XVI.  I,  =  300,  x'  =  149.9.  From  Table  IX,  T=  ^^^ 

4 

=  337.3.     Then,  bj^  (185), 

r.  =  149.9  +  337.3  +  2  62  tan  13°  15'  =  487.8  ft. 

The  station  number  of  P.T.G.  will  now  be  91.378  -   4.878  = 
86  -f  50. 


130     A    FIELD-MANUAL   FOR    RAILROAD   ENGINEERS. 


By  (186), 


E,  = 


156.7 


+  2.62  sec  13°  15'  =  41.87  ft. 


Table  XVI  gives  7,°  =  6°;  hence  the  circular  curve  -will  cover 
26°  30'  -  2  X  6°  =  14°  30',  or  3.625  stations,  so  that  the  number 
of  the  FT.,  will  be  86.50  +  (2  X  3.00  +  3.625)  =  96  +  12.5. 

157.   Tangent  Distance,  Offsets  Unequal. 

In  Fig.  71,  0,  2^,  and  K  do  not  lie  in  the  same  straight  line. 


'<- 

—X-t.-^ 

<„ _T-i J 

c 

L                         K 

9 

1 

'"— -"....^ 

--       l^      n\ 

1 

A 

->            V                 ''I     \ 

1 

• 

1 

^""^1 

1 

1 

1 

\t,y^* 

■ 

^\ 

1 
1 

/ 

/ 

\\N 

■ 

^ 

/ 

/ 

/ 

y"^ 

/                      ^ 

/                 ^ 

1               ^ 

/      ^/R 

/      ^^ 

•■ 

/    ^^ 

^ 
^ 

/  j^ 

^ 

-" 

■^ 

M 


^H 


Fig.  71. 

Draw  PS  perpendicular  to  NB,  PQ  perpendicular  to  LK.     Let 
LA  -  F,  MB  =  F'. 


or 


T   =  LK=  AN-{-NP  -  KQ, 

T' =  T-\- F' cosec  I  -  Fcoil; (187) 

T,  =  X  -{-  T'  =  x'  -\-  T-{-  F'  cosec  /-  i^'cot  /;  (188) 

T"  =  MK  =T  -  F'  cot  7+  2^  cosec  i  ;    .     .     .  (189) 

T^  =  x"  -\-  T  -  F'  cot  I  +  7^  cosec  /.....  (190) 


Example — Two  tangents  intersect  at  sta.  820  and  are  to  b(; 
united  by  a  6°  curve  having  F  -  1.75,  i^'  =  2.95,  and  /=  3r  48'. 

1632.3 


By  Table  IX,      T  = 


6 


=  272.05  ft. 


By  Table  XVI.  d  =  200,  I,  =  260,  x'  =  100,  xf'  =  129.9. 


TRANSITION-CURVES. 


131 


By  (188). 
T,  =  100  4-  273.05  +  2.95  x  cosec  31°  48'-  1.75  cot  31°  48' =  374.8. 

By  (190). 
T2  =  129.9  +  272.05 -2.95  cot  31°  48'  +  1.75  cosec  31°  48'=400.5. 

158.   To    Insert    Transition-curves    without   Changing    the 
Position  of  the  Vertex,  B. 
In  Fig.  72,  ABC  is  the  located  curve,  FGHK  the  curve  after 


Fig.  72. 

inserting  transition-curve.  The  radius  of  the  circular  portion  has 
been  changed  from  R  \o  R'  m  order  to  mal^e  room  for  the  offset 
PS=  F.  BM  =  E  is  the  external  to  located  curve,  BL  =  E' 
the  external  to  circular  curve  having  radius  R'  and  central  angle 
I.     lu  the  triangle  LNM,  LM  —  LN  sec  ^I  =  Fsec^I;  hence 


E'  =  E  -  Fsecll. 


(191) 


E  may  be  found  by  (24)  or  by  means  of  Table  IX  ;  then  E" 
becomes  known,  and  from  the  same  table  Z)'  is  found  by  dividing 
the  tabular  E  by  E'.     D'  will  be  larger  than  D, 

It  is  .sometimes  more  convenient  to  assume  D'  and  calculate  E' 
in  the  same  manner  as  E;  then,  from  (191), 


F=iE-  E')  cos  iZ. 


(192) 


If  this  value  of  i^is  too  large  or  too  small  for  the  conditions  of 
the  problem,  a  new  D'  can  be  assumed  and  7''' recomputed. 


132     A   FIELD-MAXUAL   FOR    RAILROAD    ENGINEERS. 

Example. — The  P.C.  of  a  5^  curve  is  at  sta.  182,  aud  angle 
/  =  40".  Compute  the  data  for  a  new  curve  to  allow  for  a 
transition-curve  with  1.5  ft.  offset. 

From  Table  IX,  E,  =  367.7  for  7  =  40°  ;  therefore 

E  =  ??^  =  73.54,     i^sec  20°  =  1.5  X  1.0642  =  1.6  ; 

then,  by  (191), 

E'  =  73.54  -  1.60  =  71.94, 
and 

B'  =  -^f^  =  5.1113°  =  5°  6.678',  say  5"  7'. 

By  Table  XVI,  for  h  =  200,  Z>  =  5°  7' 

F=  1.45  4-^(1.75  -  1.45)  =  1.485. 
For  li  =  220, 

F=  1.76  +  g'j)(2.11  -  1.76)  =  1.842. 

Then  for  F=  1.5,  D=  5^7', 

I,  =  200  +  20  ,  ^i~    ;  ,Q,  =  200.8  and  a-'  =  100.4. 

l.o4-&  —   1.400 

T5    n4R^         r°       ^'^      200.8  X  5.117  _  ^  ,^0       .,  ^, 
By  (146),       L    =200^ 200 -^-^^    =^   ^- 

The  central  angle  for  circular  portion  of  curve  is  40  —  2  X  5.13 
=  29.74",  equivalent  to  581.2  feet  around  curve. 

In  Fig.  72,  B  is  at  sta.  186  on  the  5°  curve,  and  arc  BG  =  290.6 
ft.  on  the  5°  7'  curve.  The  P.C.i  is  at  186  -  2.906  =  sta.  183  + 
09.4,  the  P.T.a  at  183.094-2.008  =  sta.  181  +  08.6,  the 
P.T.C.i  at  188  +  90.6.  and  the  P.  T.,  at  190  +  91.4. 

Had  D'  been  assumed  equal  to  5°  6'   or  5.1°  to  begin  with, 

we  should  have  had  E'  =  ?|^  =  72.10  ;  then,  by  (192), 

0. 1 

2<'=  1.44  X  .93969  =  1.35  ft. 

^1  may  be  found  by  interpolation  from  Table  XVI  as  above. 


TRANSITIOK-CURVES. 


133 


159.    To  Insert  Transition-curves  on  an  Existing  Road-bed 
with  the  Least  Deviation  from  Old  Track. 

To  satisfy  this  conditiou  the  new  track  should  pass  about  as 

far  outside  the  old  at  the  vertex  as  it  does  inside  at  the  original 

W 
P.O.;  that  is,  about  — .     "We  shall  now  have 


B'  =  E 


Fsec  ^-  ^. 


(193) 


The  remainder  of  the  problem  may  be  solved  by  158. 

Transition-curves  may  be  inserted  in  old  track  by  shifting  to 
suit  the  existing  road-bed,  thus  adding  materially  to  the  safety 
and  easy  riding  of  cars. 

160.  To  Insert  Transition-curves  at  the  Ends  of  a  Long 
Circular  Curve  without  Moving  the  Central  Portion. 

In  Fig.  73,  ^Cis  the  circular  curve.  In  order  to  make  room 
for  the  offset  F  the  ends  must  be  sharpened  by  compounding. 
Let  C  be  the  point  of  compounding,  li'  the  radius  of  the  branch 
ON,  HN  =  KB  =  F.      Let  BEG  be   the  transition-curve  ;   the 


B 

A 

H   • 

1 
1 

K 

R 

-^-; 

r' 

X 

L 

^'> 

'Z 

V 

0 

[? 

Fio.  73. 


closer  G  comes  to  C  the  better,  provided  the  change  in  radius  at 
G  is  kept  within  certain  limits.  The  difference  in  degrees  between 
the  original  and  the  sharpened  curve  should  never  exceed  3°  and 
may  usually  be  kept  in  the  neighborhood  of  1°. 

First  Method. — Having  decided  upon  the  value  of  F,  assume 
R  so  that  i)'  —  i)  is  not  greater  than  2°.      Draw   O'L  parallel 

OL 

to  BH;  00'  =  R-  B',  and  cos  /'  = 


00 


/> 


or 


134     A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

cos/'  -  ^-(^'  +  ^)  -  1  _  _^L  (194) 

This  is  the  same  as  (69)  in  122.  /'  being  known,  set  the  transit 
at  G,  run  out  the  curve  CN,  and  insert  transition-curve  in  the 
usual  way. 

If  /'  had  been  assumed  in  the  beginning,  R'  could  be  found 
from  (194). 

Second  Method. — "When  the  circular  curve  is  flat,  and  short 
transition-curves  are  employed,  we  ma}"  compound  the  transition- 
curve  with  the  circular  at  the  P.d,  taking  care  that  the  differ- 
euce  of  curvatures  is  not  greater  than  1°  or  2". 

Assume  the  position  of  the  P.Ci  from  100  to  200  feet  from 
the  P.C.;  measure  the  perpendicular  let  fall  from  the  P.d 
upon  the  tangent  at  the  P.C.  produced;  this  will  be  yi.  The 
central  angle  /i  can  be  calculated,  knowing  the  length  of 
circular  curve  from  the  P.C.  to  the  assumed  P.d,  or  the 
angle  between  tangents  may  be  measured  with  the  transit. 
The  coefficients  C  and  E  of  (140)  and  (142)  may  be  found 
from  Table  XIV  with  li  =  <p  as  argument  ;  then,  from  (140)  and 
(142), 

h=^^ (195) 

Xi  =  h(l  -  E) (196) 

Measure  back  from  the  foot  of  the  perpendicular  let  fall 
from  the  P.d  a  distance  a-j  along  tangent,  and  set  the  P.T.C. 

Intermediate  points  can  be  located,  if  needed,  by  offsets 
from  tungeut,  computed  by  (140)  or  (170) ;  thus  at  the  mid- 
point the  offset  is  \yi. 

Third  Method. — From  formula  (170), 


h  = 


Pi 


.005818/i' 
and,  from  (36),  yi  =  |?i^i>. 


Therefore  I.  =  ^^^^^^.  =  150  j^,  nearly. 


TRANSITION-CURVES  135 

But  nD  =  L°;  lieuce 

U  =  1507i  ; (197] 

and  as  100/i  is  the  length  of  circular  curve  from  P.C.  to  P.C.i, 
Ix  is  once  and  a  Juilf  as  great. 
From  (146), 

200V  ^  3W/ ^  2^  ^  4^_ 
li  ISO^i         150;i         3 

From  this  equation  it  is  seen  that  if  the  break  in  curvatures  is 
limited  to  2°,  this  method  is  admissible  up  to  Z)  =  6°,  independent 
of  the  length  of  transition-curve. 

Example. — A  4°  curve  is  to  have  transition-curves  inserted  at 
each  end;  compute  the  necessary  data. 

By  First  Method. — Assume  a  1.45-ft.  offset,  and  the  curva- 
ture to  be  changed  from  4°  to  5°  by  compounding.  In  Table  I 
find  R  =  1433.7,  E'  =  1146.3;  then,  by  (194), 

1  45 
cos  1=1-  -1^  =  .99494  =  cos  5°  46'. 

286.4 

5  767 
The  length  of  5°  curve  is  -^-- —  =  1.153  stations,  and,  by  Table 

o 

XVI,  /i°  =  5°,  so  that  the  P.d  will  fall  15.3  ft.  back  of  the 

5  767 

P.aC,  while  the  P.C.  will  be  moved  forward  -— 1.153  = 

4 

.289  stations  or  28.9  ft. ;  the  P.T.C.  being,  by  Table  XVI,  100  ft. 

back  of  the  new  P.C.  will  fall  100  -  28.9  =  71.1  ft.  back  of  old 

P.C.     The  transition-curve  may  now  be  located  in   the  usual 

manner. 

By  Second  Method, — Assume  the  P.d  to  fall  150  ft.  from 
the  P.C,  making  I,°  =  1.5  X  4  =  6°.  From  Table  XV,  C  = 
.03488,  and,  by  (36),  y,  =  1(1.5)^  X  4  =  7.875  ft. 

By  (195), 


Now,  by  (146), 


i)'  X  225.8 
"  "      200       ' 


136     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

from  which  D'  =  5.314°  =  5°  18.8',  which  differs  less  than  2" 
from  D. 
By  (196), 

a-,  =  225.8(1  -  .0011)  =  225.6  ft. 

To  find  the  position  of  P.T.C.  with  reference  to  the  old  P.O. 

consider  that  the  distance  from  P.O.  to  foot  of  perpendicular  from 

the  P.C  1  is  half  the  chord  for  augle  2/i ,  and  can  be  taken  from 

1      1197  9 
Table  IX.  being  equal  to  -  X  —r-  =  149.7.   Then  225.6  -  149.7 

=  75.9  feet  is  the  distance  from  old  P.C.  back  to  P.T.C. 

By  Third  Method.— Assume  the  P.  C.i  to  be  150  ft.  from  the 
old  P.  C. ;  then,  by  (197),  U  —  225  ft.,  and,  by  (198),  the  curvature 
of  transition-curve  at  the  P.d  is  |  X  4°  =  5°  20',  giving  almost 
the  same  results  as  by  the  second  method.  Had  we  taken  the 
P.C,  160  ft.  from  P.C.  we  should  have  had  U  =  240,  D'  =  5°  20'; 
iCi  =  239.7,  by  interpolation  from  Table  XVI;  the  length  along 
tangent  from  P.C.  to  foot  of  perpendicular  from  P.d  159.9  ft., 
and  therefore  239.7  -  159.9  =  79.8  ft.  as  the  distance  from  P.C. 
to  P.T.C. 

161.  To  Insert  Transition-curves  at  the  P.C.  and  P.C.C.  of 
a  Compound  Curve  by  Changing  the  Curvatures  of  the  First 
Branch. 

In  Fig.  74  let  ABV  be  the  located  curve  compounding  at  ^. 
Two  cases  occur. 

First  Case. — Second  branch  having  shorter  radius. 

The  offset  at  P.C.C.  must  be  to  outside  of  located  curves;  let  it 
be  EB  =  F^  in  the  figure.     Let  CP  =  Fbe  known  or  assumed. 

Draw  the  tangent  BG,  and  draw  EH  parallel  thereto.  Let  CE 
be  the  changed  curve,  and  CQ  parallel  to  tangent  AH.  Angle  / 
may  be  computed  from  the  known  station  numbers  of  A  and  B, 
or  may  be  measured  on  the  ground.  The  new  tangent  distance  is 
EQ  =  BG  -  OK-  HQ  (or  LS).  From  the  right  triangle  GHK, 
GK  =  HK  tan  GHK  =  F^  cot  /. 

Similarly,  LS  =  LW cosec  I  =  F cosec  /.     Therefore 

T'  =  EQ=  T  -  F^cot  I  -  Fcosec  I.     .     .     (199) 

T  can  be  found  from  Table  IX  or  formula  (14);  then  T'  is 
known  from  (199).  The  degree  of  new  curve,  Z>',  may  now  be 
found  by  means  of  Table  IX,  or  from  Table  I  by  first  finding 


TRANSITION-CURVES. 


137 


R'  by  (15)      The  transilion  curve  at  the  P.  C.C.  may  be  located  by 
146  aud  151,  while  that  at  the  P.C.  may  be  located  either  by 
offsets  or  deflectious. 
Second  Case. — Second  branch  having  longer  radius. 


Fia.-  74. 


In  this  case  the  offset  must  be  to  inside  of  curve,  and  IfS  is  the 
tangent  required.     From  the  figure,  letting  NB  =  F^,  NS  —  T', 


T  -  r+  F^  cot  I  -  i^cosec  /. 


(200) 


The  remainder  of  the  solution  is  the  same  as  for  first  case. 

Example. — A  5°  curve  compounds  at  sta.  280  with  a  9°  curve; 
the  P.C.  is  at  sta.  272.  Required  the  change  in  curvature  of 
first  branch  for  an  offset  of  1.50  ft.  at  F.C.  C.  and  2.00  ft.  at  P.O. 

Here  7=8x5°^  40°,  and,  by  (199), 


T'  =  417.1  -  1.5  X  1.19175  -  2  X  1.5557  =  412.2 

2085.5 


D'  = 


412.2 


=  5.06°  =  5°  3.6'. 


Da  —  ZX  =  9  —  5.06  =  3.94°  is  the  difference  in  curvatures  of 
the  two  branches  of  the  altered  curve.  Entering  Table  XVI 
with  this  value  for  D  and  F  =  1.5  ft.,  we  find  : 


138     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


for  li  =  220, 
*'   lx=  240, 


F=  1.06+  .94(1.41  -  1.06)  =  1.39; 
F=  1.26+  .94(1.67  -  1.26)  =  1.65. 


1    K  1    QQ 

.-.  for  F=  1.5,  D  =  3.94%     h  =  220  +  20^  _,       "I-  =  228.5 

1  .bo  —  1.  0«7 

Bisect  the  offset  at  P.C.C,  measure  114.25  ft.  along  each  curve, 
aud  set  the  ends  of  transition-curve.  Midway  between  these 
points  and  P. CO.  offset  -^^  X  1.5  =  0.1  ft.;  these  are  all  the 
points  needed. 

The  length  of  transition-curve  at  P.O.  may  be  found  in  like 
manner,  taking  Z)  =  5.06°  and  j^=2.0  ft.  as  arguments  in 
interpolating  in  Table  XVI. 

162.  To  Insert  Transition-curves  at  the  Ends  of  Two  Circu- 
lar Curves  of  Contrary  Flexure  united  by  a  Common  Tangent. 

In  Fig.  75  let  the  located  line  be  ABCE;  the  tangent  BC  must 
be  shifted  outward  at  B  and  C  to  the  position  UG ,  the  relative 
size  of  offsets  being  determined  by  the  nature  of  the  ground. 
The  points  B'  and  C  at  which  the  tangents  to  circular  curves 


r 

ii     ; 
I 

Fig.  75. 

will  be  parallel  to  HQ  will  each  move  towards  S  a  distance  due 
to  the  increase  of  central  angle,  which  increase  equals  B8H=  a, 
for  which  we  have 

tan  a — (201) 

Let  the  offset  at  B'  be  F,  and  at  C",  F'.     Then 

F=  (R-\-  BH)  cos  a  -  B.    .     .     .     .     (202) 
F'  =  (A"  +  CG)  cosa  —  E' (203] 


TRANSITION-CURVES.  139 

F  aud  F',  being  now  known,  the  transition-curves  may  be 
located. 

Example. — A  G°  curve  and  a  4°  curve  are  united  by  a  tangent 
540  ft.  long;  57/ for  G^  curve  =  4.5  ft.;  CO  for  4°  curve  =  3  ft.; 
B  is  at  sta.  180,  G  at  185  +  40.     Find  F  and  F'. 

By  (201),  tan  a  =  ^^  =  .0139  =  tan  0°  48'. 

o 

B  will  be  moved   forward  '     =  .133  stas.  =  13.3  ft.  to  sta. 

6 

a 

180  +  13,3,  and  G  will  be  moved  backwards  '—  =  .2  stas.  or  20 

ft.  to  185  +  20. 
By  (202),        F=  (955.4  +  4.5)0.99990  -  955.4  =  4.4  ft. 
By  (203),       F'  =  (1432.7  +  3)0.99990  -  1432.7  =  2.86  ft. 

These  values  call  for  ly  =  317.8  ft.  for  6"  curve,  and  h  -  313.3 
for  4°  curve. 

Remark. — It  will  frequently  be  found  that  this  problem 
allows  the  line  to  be  thrown  on  better  ground.  Should  the 
ground  require  tangent  to  be  shifted  inward,  the  curves  must 
be  sharpened  by  compounding  to  admit  of  the  necessary  offsets. 

163.  Having  Run  a  Tangent  which  Falls  Outside  a  Located 
Curve,  to  Find  the  Offset  F  for  a  Transition-curve  Uniting 
them. 

lu  Fig.  76  let  the  tangent  be  AB  ;  CF  the  located  curve.  Set 
transit  at  some  point  C,  and  bring  telescope  into  tangent  to 
curve.  Measure  CB  and  move  to  , 
B,  where  angle  ABC  must  be 
measured  ;  or  measure  CH  per- 
pendicular to  AB ;  then 

CII 
sm  a  =  — . 

Now  FG  =  R  vers  a  ;  or  it  is 
the  mid-ordinate  for  twice  a,  and 
may  be  found  from  Table  IX ; 
then 

F=CR-EO=  CH-nveraa.     .     .     .    (204) 

The  point  E  is  found  from  C  by  the  relation  EC  =  jz- 
The  transition-curve  may  now  be  located. 


140     A    FIELD-MANUAL   FOR    RAILROAD   ENGINEERS. 

164.  Inserting  Transition-curves  in  Old  Track. — Sections 
159  aud  160  afford  the  means  of  inserting  transition-curves,  of 
which  159  is  theoretically  the  best,  though  from  the  amount  of 
track  disturbed  it  may  be  better  to  employ  160.  Sometimes  the 
method  of  162  may  be  employed  to  advantage  when  the  connect- 
ing tangent  is  short.  For  easing  the  curves  at  point  of  com- 
pounding, the  method  of  161  may  be  made  use  of. 

The  offsets  must  necessarily  be  small  if  the  new  track  is  re- 
quired to  occupy  the  old  road-bed.  It  may  be  profitable  to  add 
to  the  road-bed  when  sufficient  offset  cannot  be  secured  for  sharp 
curves,  though  ordinarily  much  good  can  be  accomplished  even 
when  the  new  track  is  restricted  to  the  old  road-bed. 

Unless  the  theoretical  P.C.,  P.C.C.,  and  P.T.  have  been 
marked  by  monuments  it  may  be  diflicult  to  retrace  the  old 
lines.  If  there  is  plenty  of  room,  the  terminal  tangents  may  be 
prolonged  to  intersection  and  /measured,  after  which  the  degree 
of  curve  may  be  found  by  measuring  around  curve  aud  by  ap- 
proximate measurements  of  the  tangent  distances  ;  then  one  or 
two  assumptions  and  computations  will  generally  suffice. 

In  cuts  and  rough  country  the  curve  may  be  run  out  by  setting 
transit  in  center  of  road-bed  and  measuring  the  deflection-anglea 
for  a  few  points  around  the  curve. 

After  the  transition-curves  have  been  inserted  permanent  monu- 
ments should  be  placed  at  each  end  of  transition-curve  to  guide 
the  trackman  in  keeping  up  the  proper  superelevation  of  oute' 
rail. 

165.  Remarks  on  Tabular  Interpolations. — The  general  inter 
polation  formula  given  in  algebra  is 

in  which  t  is  any  term,  a  the  first  term  taken,  p  the  number  of; 
terms  from  a  to  t,  fZi  the  first  from  a  of  the  first  order  of  differ 
ences,  do  the  first  of  the  second  order  of  differences,  etc. 

In  ordinary  linear  interpolation  all  terms  after  the  second  are 
neglected  ;   in  interpolating  by  second  differences  all  after  the 

third,  etc. 
In  Table  XIV  linear  interpolation  will  answer  for  C  and  ordi- 


TRANSITION-CURVES.  141 

narily  for  ^.though  second  dilBt'erences  may  sometimes  be  needed 
for  ihe  latter. 

In  Table  XV,  A  is  a  quadratic  function  of  n,  as  shown  by  for- 
mula (15;i),  while  5  is  a  cubic  function  of  that  portion  that  lias 
been  retained.  Hence  A  should  be  interpolated  by  second  differ- 
ences, while  theoretically  B  should  be  interpolated  by  third 
differences  ;  but  as  B  is  always  quite  small,  its  second  and  third 
differences  will  be  too  small  to  affect  results,  and  linear  interpola- 
tions may  be  made  when  any  are  needed. 

In  Table  XVI  linear  interpolations  will  generally  suffice, 
though  when  ii^and  y  are  large  it  may  be  necessary  to  use  second 
differences. 

The  examples  of  158  and  161  illustrate  the  method  of  inter- 
polating in  Table  XVI  for  intermediate  values  of  F  and  D 
Values  of  7^  were  first  found  for  the  given  degree  of  curve  and 
assumed  values  of  ^i ,  so  taken  that  the  true  ^i  should  be  between 
them.  From  these  assumed  values  of  Ij  and  F,  taken  with  the 
required  F,  the  true  h  was  found  by  linear  interpolation. 

As  an  extreme  case  suppose  F  and  2/1  wanted  for  an  18^  curve 
when  li  =  408  feet. 

First  write  a  few  values  of  ^i  and  F  so  as  to  obtain  the  first 
and  second  differences. 

Z,  y,  rfi  d^  F  rf,  ^2 


400  81.43  20.63 

8.08  2.08 

420  89.51  0.35  22.71  0.10 

8.43  2.18 

440  97.94  0.35  24.89  0.10 

8.78  2.28 

460        106.72  27.17 


By  the  interpolation  formula,  when  ?i  =  408, 

y,  =  81.43  +  is  X  8.08  +  4^_zi.)  X  0.35  =  84.62, 


F  =  20.63  +  ^%  X  2.08  +  '^^^, — -  X  0.10  =  21, 
By  linear  interpolation,  .y,  =  84.60,  F=  21.40. 


45. 


142     A   FIELD-MAXUAL   FOR    RAILROAD    EXGINEERS. 
Again,  suppose  ^i  to  be  wanted  when  h  —  430.  By  the  formula, 

yy  =  81.43  +  M  X  8.08  +  IMLzii-^  X  0.35  =  93.68, 


or 


y,  =  89.51  4-  i^  X  8.43  +  ^^^ — ^-  X  0.35  =  93.68. 


CHAPTER  V. 

*^  FROGS  AND  SWITCHES. 

Article  ^4,    Turnouts. 

A,   Turnouts  from  Straight  Lines. 

166.  A  Turnout  is  a  track  used  in  leaving  the  main  line.  A 
Frog  is  placed  at  the  intersection  of  main  and  turnout  rails. 

a.  The  Gauge-line  is  taken  as  coinciding  with  inside  face  of 
rail.  In  makiug  measurements  between  tracks  the  distance  be- 
tween corresponding  gauge-lines  is  what  is  wanted. 

b.  The  Gauge  of  track  is  the  distance  between  gauge-lines  of 
the  rails  of  that  track. 

c.  The  Point  of  Switch  is  the  point  at  which  the  turnout  curve 
begins  ;  for  a  point  switch  (split  switch)  this  is  at  the  head-block, 
while  with  a  stub  switch  it  is  the  length  of  the  switch-rail  back 
of  the  head-block,  which  is  at  the  toe  of  switch. 

d.  The  Frog-point  is  at  the  intersection  of  the  gauge-lines  of 
intersecting  rails,  and  lies  a  few  inches  in  front  of  the  blunt 
point  of  frog  as  manufactured. 

The  angle  formed  by  the  intersecting  gauge-lines  is  the  Frog- 
angle. 

e.  The  Frog-number,  J^,  is  the  ratio  of  the  axial  length  to  the 
width  of  base  of  frog. 


In  Fig.  77, 


Fig.  77. 


AE       k 


CB~  w 


143 


144     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 


Letting  the  frog-angle  BAG  ha  F,  the  figure  yields 

(205) 

(206) 


^  ^      ^w        1 
cot  \F=  ^  =  2N. 


f.  The  Lead,  I,  is  the  distance  from  point  of  switch  to  point  of 
frog,  measured  along  that  main  rail  in  which  the  frog  is  placed. 
In  Fig.  78,  CB  =  I. 

g.  The  Stub-lead,  s.l.,  is  the  distance  along  main  rail  from  frog- 
point  back  to  a  point  where  the  turnout  rail  diverges  from  main 
rail  an  amount  equal  to  the  throw.  In  Fig.  78,  KB  =  s.l.  =  I  — 
length  of  switch-rail. 

7i.  The  Throw,  i,  of  switch-rail  is  the  distance  the  point  of  a 
split  switch,  or  toe  of  stub  switch,  is  moved  in  opening  or  closing 
the  switch.  A  distance  of  from  5  to  5f  inches  is  needed  to  give 
necessary  clearance  for  flanges. 

k.  The  Frog-distance,  f.d.,  is  the  length  of  the  chord  of  outev 
rail  of  turnout  from  the  point  of  a  split  switch,  or  toe  of  stub 
switch,  to  the  point  of  frog. 


167.  Given  the  Frog-number,  iV",  and  the  Gauge,  g,  of  a 
Turnout  from  a  Straight  Line,  to  Find  the  Lead,  I,  and  Radius, 
M,  of  Center  Line  of  Turnout. 

A  G     E 


Fig.  78. 

In  Fig.  IS,  AC  =  g,  CB  =  I,  angle  ABC 
From  the  figure,  I  =  g  cot  iF. 

But,  (206^  cot  iF  =  2N. 

.-.     I  =  2glf.     . 


=  \F. 


.     (207) 


FROGS   AND    SWITCHES.  145 

From  triangle  OBG, 

(H  +  kf  -  (^  -  iff)"  =  ^  =  ^9'N\ 

whence  2gR  =  4g^N\ 

.-.     R  =  2gN''  =  IN. (208) 

5730 
Taking  22  =  — yy-,  inserting  in  (208),  and  solving  for  i), 

-=S <-> 

For  g  =  4iii.  8^  in.  these  formulas  become 

;  =  9.42iV  feet, (207') 

72  =  9.42iV^'' feet, (208') 

2>  =  ^  degrees; (209') 

and  for^  =  4  ft.  9  in., 

I  =  9.5iV, (207") 

R  =  9.5N\ (208") 

i>=|| (209") 

If  the  frog-distance  AB  is  wanted,  we  have  AB  =  |/Z*  +  ^',  or 
f.d.  =  g  ViiV^TT  =  9  cosec  IF.      ...    (210) 

Example. — Find  I,  R,  D,  and  f.d.   for  a  No.  8  frog  and  4.75 
feet  gauge. 

By  (207").  Z  =  9.5  X  8  =  76.0  ft.; 

"   (208"),  i?  =  9.5  X  64  =  608  ft.; 

"   (209"),  D=^  =  r  25'; 

64 


"  (210).  f.d.  =  4.75  t/257  =  76.14  ft. 


146      A    FIELI>-MA?^UAL    FOR    RAILROAD    ENGINEERS. 

168.   Given  7?  (or  D)  and  g,  to  Find  N,  I,  and  F. 
From  (208)  and  (209), 


,  /R        ,  /5730 


53.52 


2g       y    2gD        ^^-     ' 
From  (207)  aud  (211), 

l=2gN  =  2g\/^  =  |/2^  =  107  Y^. 
From  (206)  and  (211), 

cot|^.2i.=  2/|=|/f  = 


(211) 


107 


2ff  ^      g  \/gB 

F  m&y  also  be  found  from  triangle  OBC.  Fig.  78  : 

cos7^=4=^ 


(212) 


(213) 


(214) 


169.  To  Find  the  Length  of  Switch-rail,  S,  when  the  Frog- 
number,  3",  the  Throw  of  Switch,  t,  and  the  Gauge,  g,  are 
Given. 

In  Fig.  78,  by  geometry, 

HG  ^' 


2(i?  4-  Iff)  +  HG' 
Neglecting  the  HG  in  denominator  as  small, 


JIG=       -^» 


In  like  manner,      KL  = 


2(i?  +  yy 

2{R  -  W 


Writing  AG  =  Aff=  CK  —  S,  and  taking  the  mean  of  de- 
nominators, 

^-  2R' 


whence  S  =  y^Rt  =  2JV  Vgi:      ...     *  (215) 

wf       7?      5730 

Writmg  R  =  -^,  , 


8=  a/  2t^^=   107  |/- (216) 


FROGS   AND    SWITCHES. 


14' 


170.  Given  the  Main  Frog-number,  N,  to  Find  the  Num- 
ber, Ni  .  and  Lead,  ^,  ,  of  Crotch-frog  for  a  Turnout  from  Both 
Sides  of  Straight  Main  Track. 

Ill   triangle    OCII,   Fig.    79,  re- 
memberiug  that  E  =  2gN^, 


cos  ^Fi  =  — 


R 


AN^ 


(217) 


A' +  1(7    AN'+l 
Tlfeea,  by  (206), 

iv^, =icotii^,..  .  .  (218)  |_.4^_7^;:r:^^c^'i 

From  the  figure  and  (205),  -^ — ^ 

I,  ^  Rtau  IF,  =^.=g^^;   (219) 
also. 


=  ^r  |/2iV"M^~i.  ....     (220) 


Equating  these  values  of  li  and  solving  for  Ni  gives 


N, 


V2iV"^+i 


(221) 


If  the  \  in  denominator  be  neglected  as  small  compared  "with 
2N\  (2C1)  becomes 

N,  =  -^=  0.707iV: (222) 

|/2 

If  in  (220)  we  neglect  the  |  under  radical,  tbere^results 

i,  =  g]^  i/2  =  lAUgN  =  O.IOIl.     .     .     .     (223) 

The  distance  between  main  and  (Totch  frogs  measured  along 
main  rail  is 


I-  I,  =2gN-  g  V2JV'+i (224) 

or,  approximately, 

I  -Uz=  2gN  -  \AUgN  =  O.^SGgJST  =  0.293i.    .     (225) . 


148     A    FIELD-MANUAL    FOR   RAILROAD   ENGINEERS. 


171.   To  Find  the  Radius,  R,  of  Turnout   and  Lead,  ^i ,  of 
Orotch-frog  in  Terms  of  the  Crotch-frog  Number,  Ni 


From  (222). 


iV^^  =  2N^\ 


Insert  this  in  (208)  and  (219),  giving 

E=2g.  2N^-'  =  4gNx\ 


(226) 


h  =  ^-^^  =  2gN, (227) 

Remark.— In  general  the  frogs  kept  in  stock  by  manufacturers 
do  not  afford  suitable  combiuatious  of  numbers  for  double  turn- 
outs. For  instance,  the  theoretical  number  of  crotch-frog  for  a 
number  8  main  frog  is,  by  (221)  or  (222),  iVi  =  5.66,  and  we  should 
be  compelled  to  use  a  number  5^  or  6  for  the  crotch-frog;  this 
would  necessitate  a  different  rate  of  curvature  from  crotch  to 
main  frog  than  from  head-block  to  crotch. 

172.  Given  the  Numbers  of  Middle  Frog,  iV,  ,  and  of  Main 
Frogs,  iVand  N',  to  Find  the  Radii  li^  from  Point  of  Switch 
to  Crotch-frog,  and  R  and  M',  from  Crotch  to  Main  Frogs. 

In  Fig.  80  we  have,  by  (226), 

0,N=B,=4gNx\ 

and,  by  (227), 

NO  =U=  2gN,. 

Now  if  Fx ,  F,  and  F'  are  the  an- 
gles of  the  frogs  Ni ,  JV,  and  N', 
the  angle 

COH  =  F-  ^F, , 
and 


%  CHQ=F-l{F-\F,)  =  \{F+\F,). 


Since  CG  =  \g,  the  triangle  GEO 
yields 

OH  =  \g  cot  \{F  +  ^F,).   .     (228) 

But,  by  trigonometry, 


..t  (xw-x-  XW^  -  l-taniF.tanjJ^. 
cot  (ii^+  IF.)  -  -_-^^^-— p^ 


FROGS   AND   SWITCHES.  149 

Assume  tan  ^Fi  =  ^  tau  |F, ,  aud  write 

and  after  simplifying  and  reducing, 

The  last  term  is  (juite  small,  rarely  amounting  to  as  much  as 
one  inch,  aud  may  be  neglected  ;  then 

2gNN,     _       hlf      _       m, 

From  the  triangles  LCO  and  KHO, 

(B  +  Ig)  cos  IF,-{H  -f  \g)  cos  ii^  =  \g, 
whence 

In  like  manner  for  the  curve  CE, 

ME  =  y  cot  K^'  +  i^O  =  ^/^  _^  ^,.    .     .     (232) 

i2'  •+  1^^  = -~-^- —        .     .     .     (363) 

-^       2(cos  ^Fi  —  cos  F  ) 

Example. — Given  iV",  =  6,  iV^=  8,  and  N'  =  9,  to  find  the  lead 
li,  the  distances  GH  and  ME,  and  radii  R,  Ri ,  and  R',  g  being 

4.75  ft. 

By  (226),    i?i  =  19  X  6^  =  684  ft.,  an  8°  23'  curve. 
By  (227),      ^1  =  9.5  X  6  =  57  ft. 

By  (230).   GH  =  — ^f^^^  =  32.8  ft. 
By  (232),  ME  =  24.4  feet. 


150     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


By  (231),      R  = 
a  10°  28'  curve. 

By  (233),    li' = 
a  6°  32f  curve. 


4.75 


2(cos  4°  46'  -  cos  7"  9) 


2.38  =  547.4  ft., 


4.75 


2(cos  4°  46'  -  cos  6°  22  ) 


-  2.38  =  876.4  ft. 


173.  Given  the  Number,  iV^,  of  the  Two  Main  Frogs  and  the 
Gauge,  g,  to  find  the  Crotch-frog  Number,  iV, ,  its  Lead,  li  ,  and 
the  Radius,  lii  ,  of  Curve  through  Crotch  when  the  Double 
Turnout  is  to  Same  Side  of  Straight  Main  Track. 

In  Fig.  81  the  frogs  at  B  and  G  are  of  the  same  number,  and 
may  be  taken  as  falling  on  the  same  straight  line  through  the 
center  0.  Angle  0,G0  =  90°  -  OGL  =  F,  and  the  triangle 
00i(r  is  therefore  isosceles;  hence 


OiG  =  0,0  =  OA  -  0,0  =  lOA, 


or 

whence 


(234) 


Now,  by  the  same  reasoning  as  in  167,  2gNi'^  ~  R, ,  whence 


^.  =  /f:= 


'R 

4g 


(235) 


FROGS   AND    SWITCHES. 


151 


Neglecting  the  J  under  radical  and  willing  B  =  2gN'  gives 

N,  =  ^_  =  .707iV, (236) 

1/3 

which  is  identical  with  (222)  for  turnouts  to  opposite  sides.     For 
^Cand  EB,  as  in  167,  U  =  'igN,  and  I  =  2gN.     Hence 

GB=  I-  I,  =  2g{]Y-  J^J (287) 

Example. — Find  Wi,  i?i ,  and  l  —  li  where  iV^  =  9  and 
g  =  4.75  ft. 

By  (208"),     ^    =  9.5  X  81  =  769.5  ft.,  a  7°  27'  curve. 

"  (234),      Bi   =  384.75  -  1.19  =  383.56  ft.,  a  14"  56'  curve. 

"   (236),      N,  =  .707  X  9  =  6.36. 

"  (237),       GB  =  1  -  h  =9.5x  3.64  =  25.08  ft. 

Remark. — It  may  now  be  seen  that  the  proper  combination  of 
frogs  for  a  double  turnout  to  opposite  sides  applies  also  where 
the  turnouts  are  to  same  side  of  straight  main  line.  Also  they 
apply  to  turnouts  from  opposite  sides  of  curved  main  line  when  its 
radius  is  not  less  than  that  required  by  main  frog  for  straight 
track. 

174.  Given  the  Number  of  Main  Frogs,  iV",  and  of  Crotch- 
frog,  Ni ,  to  Find  the  Radius  of  Curve  between  Frog-points  of 
a  Double  Turnout  to  Same  Side  of  Straight  Track. 


0 

Fig.  82. 
In  Fig.  83,  O-iG  -  B^-\-  \g,  and  the  chord  CO  must  be  deter- 
mined.    The   frogs  at   B  and    G   being  of  the    same  number, 
0,00  =  GOOi  =  F  and   CO,E  =  F,. 


152     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


Draw  (?^  perpendicular  to  EB;  then  in  triangle  BGH 

QH  =  g  cos  F. 

Draw  O^L  perpendicular  and  (?Z  parallel  to  EB\  from  tri- 
angles OaG^-K'and  O^CL, 

(ij,  ^  ^^)(cos  Fi  -  cos  'iF)  =  KL  -  GH  =  g  cos  F, 


whence 


It2  +  hg  = 


g  cos  F 


(238) 


(239) 


cosi^'i  -cos2i^"    •     * 
From  triangle  O^CQ,  since  GO^G  =  2F  -  Fi, 

CG  =  2{R,  +  ^g)  sin  U^F  -  F,).  .     . 

Example.— Given  N  =  S,  JSTi  =6,  and  g  =  4.75,  to  locate  the 
turnout. 
By  (208),  R  =  608  ft. ;  Ri  =  342  ft. 

By  (238),  Bi  +  y  =  274.5  ft. 

By  (239),  CG  =  22.8  ft. 

175.  Given  the  Frog-number,  N,  the  Gauge,  g,  and  Distance, 
p,  between  Centers,  to  Unite  Main  Line  with  a  Parallel  Siding 
when  the  Reversing-point  is  at  Frog-point. 


In  Fig.  83,  BOx  =  Ri  -  \g  and  BE  are  required. 
In   triangle  50,^,  BO,  =  R,  -  \9,  EO,  -  R,  +  ^g  ~  p,  and 
angle  BO,E  =  F.     By  trigonometry, 


FROGS   AND   SWITCHES.  153 

(Br  -  y)  +  (Jg.  +  y-p)  ^  tan  i(180°  -  F) 
(i?i  -  ig)  -  (Ri  +  y  -P)  tau  ^F 

2R^p^cotiF 
p  -  g        tan  ^F 

whence  Ei  .=  2(p  -  g)N''  -\- \p, (240) 

BE  =  (i?x  -ig)smF. (241) 

T>7f  7 

From  the  similar  triangles  ABG  and  BCE,  =  - ,  from 

p-g      g 


t   • 


which 


BE  =  ^P^^  =  (?  -  l]l.    ....    (242) 


g-l).    .     .    .    . 


Example.— Find  Bi  and  BE  when  iV  =  8,  p  =  12.35  ft., 
g  =  4.75  ft. 

By  (240),      El  =  15.2  X  64  +  6.2  =  979  ft.,  a  5°  51'  curve. 

By  (207").        i  =  9.5  X  8  =  76  ft. 

By  (242),     BE  =  (2.6  -  l),x  76  =  121.6  ft. 

Remark. — If  space  requires  that  the  turnout  get  away  from 
main  line  more  rapidly  than  by  the  above  method,  we  can  assume 
the  second  radius  equal  to  or  less  than  the  radius  of  turnout  and 
find  the  reversing-point  by  131,  and  then  compute  BE. 

176.  To  Lay  Out  a  "  Ladder"  Track 

In  yardwork  a  number  of  parallel  sidings  may  be  conveniently 
connected  with  the  main  line  by  means  of  a  ladder-track. 

In  Fig.  84,  if  the  frog-number  iV  and  the  distance  p  between 
center  lines  of  track  are  given,  it  is  only  necessary  to  determine 
the  distances  BC,  CE,  etc.,  between  frog-points,  and  BK,  CL, 
etc.,  between  point  of  switch  and  point  of  frog.  From  triangle 
BCQ, 

BC  =  -r^  =  p  cosec  F, (243) 

sm  ^ 

BK=  BO  -  KG  =  p  cosec  F  -  2gN ;  .     .     .     (244) 


154     A    FIELD-MAKL'AL    FOR   RAILROAD    ENGINEERS. 


or,  since  cosec  F  —  iV-|-— r^,  (see  186,) 


BG  =pN-\- 


(243') 


BK=  (p-2g)N  + 


V 


(244') 


Example.— For  a  No.  8  frog  find  5Cand  BKvi\ien  p  =  12.8  ft. 
and^  =  4.75  ft. 

By  (243),  BC  =  12.8  X  8  +  -^  =  102.8  ft. 


By  (244), 


BK=    3.3  X  8  +  0.4  =  26.8  ft. 


B.    Turnouts  from  Curves. 

177.  Given  the  Radius  of  Main  Curve,  the  Frog-number, 
and  the  Gauge,  to  Find  the  Radius  and  Lead  of  Tvurnout  from 
Concave  Side  of  Main  Line. 

In  Fig.  85,  AB  is  tlie  outer  rail  of  turnout,  CB  the  inner  rail  of 
main  track.     In  triangle  OAB,  since  O2BA  =  GAB, 

OBA  -  OAB  =  F, 
OB  A  +  OAB  =  180'^  -  0, 
and     OA  ^  R+  ^_g.   OB  =  R  -  Ig. 


FROGS   AND   SWITCHES. 


155 


Then,  by  trigonometry, 

(i?  +  ha)  -\-iIi  -  jg)  _  tan  ^(180  -  6) 
{R  +  \g)  -  (H-  W  ~        tan  ^F 


cot^e 
tan  hF' 


9  7?  7? 

Reducing,       cot  |9  =  y  ^an  ^2^  =  ^. 


Then 


I  =  BC  =  2{R  -  y)  sin  49.       . 


(245) 
(246) 


If  the  length  of  AB  is  wanted,  we  can  sliow  that  the  angle 
ABC  =  \F;  and  by  solving   the  triangle  ABC,  since  ACB  = 

90°  +  i/, 

g  cos  ^0 


AB 


sinii^ 


(247) 


To  find  Ri ,  from  triangle  dAB, 


2(.ff2  +  Ig)  sin  liF  +  6)  =  ^5. 


(248) 


Or,  in  triangle  BO^C, 


{R.  +_i£)_+  {R-^-_hu)  ^  tan  i[180  -  (/^+  «)] 

(/?:;+  is')  -  (^^2  -  \g) 


tan  ^i^ 


coti(F+e) 

tan  \F      ' 


156     A    FIELD-MANUAL   FOR    RAILROAD    ENGINP^ERS. 
Reducing  aud  solving  for  i?,, 

^'2'      tan  ^F      =  ^  •  ^«^  i^'-  cot  K^+  fi).  .    (249) 
But,  from  trigonometry 

cot  i(^+ 6, = cot  ,iF+ m ='  :jpf,::]t- 

Substitute  this  in  (249)  and  write 


cot  ^F  =  2]Sr,    tan  ^F  =  ^r^,     tan  ^0  = 


—       tan  iQ  -  ^^ 


and  reduce ;  then 


For  2^iV*  write  i?i ,  the  radius  of  turnout  from  straight  track, 
and  neglect  the  ^g  in  numerator  as  small  compared  with  B;  then 


«'  =  4^ <2«» 


Now  write 


5730      „  5730       „         5730 


and  reduce,  yielding 

Da  =  D  +  D, (253) 

Formula  (252)  affords  an  easy  method  of  finding  the  degree  of 
turnout  curve,  or,  if  preferred,  the  radius  may  be  first  found  by 
(251). 

Draw  Oi?to  the  mid-point  of  CB  ;  OE  does  not  differ  greatly 
from  OB  or  OC ;  so,  if  we  write  OE  =  R  —  ^g,  there  results 

aJSf  a'^N 

1  =  2{B-  Ig)  tan  \0  =  2{R  -  lgf~  =  2gN  -  V-     (253) 

The  last  term  is  quite  small,  even  in  the  most  extreme  case 
likely  to  arise  in  practice  ;  for  a  turnout  from  a  6°  curve  with 


FROGS   AND    SWITCHES. 


157 


number  8  frog  it  nmoimts  to  only  2|  inches  ;  neglecting  it,  we 
may  write,  as  for  straight  main  track, 


I  =  2gN. 


(254) 


Example. — Turnout  from  inside  of  a  4"  curve,  ]V=S,  </  = 
4.75  ft. 

By  (208"),  Bi  =  9.5  X  64  =  COS  ft.,  a  9°  26'  curve. 

By  (252),    i>2  =  4°  +  9^  26'  =  13"  26',  for  which  E^  =  426.8. 

By  (254),       I  =  76  ft. 

178.  Given  the  Frog-number,  the  Gauge  and  Radius  of  Main 
Curve,  to  Find  the  Lead  and  Radius  (or  Degree)  of  Turnout 
from  Convex  Side  of  Main  Line. 

In  triangle  AOB  of  Fig.  86,  ^  +  ^  =  180°  -  6, 

A  -B  =  (180°  -  O^AB) 

-{ISO" -O^B  A- F)  =  F. 

Bj'^  trigonometry, 
ifi^W^-j-  {R  -  Ig)      tan  |(180  -0) 


2R 

9 


tan  IF 


or 
whence 


cot  IB 
tan  AF 


27? 


cot  IB=  ~  tan  IF= 


R 


(255) 


9  -        9N 

From  triangle  OCB, 

l=CB  =  2{R  +  Ig)  sin  |(3.   (256) 
Assuming  OE  —  R  ^^  \9, 
i  =  2iR  +  Ig)  tan  10 

Neglecting  the  last  term  as  small,  as  in  177, 

l  =  2gN,,     .     .     . 


which  is  the  same  as  (207). 
In  triangle  CO,B,  0^  =  F  -  0. 


(258) 


•.  tauiOa  =  tan  {IF  -  |0). 


158     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

We  may  now  follow  the  same  Hue  of  reasouing  by  which  (231) 

was  derived,  or  more   simply  by  assuming  the   tangent   of   the 

difference  of  two  small  angles  equal  to  the  difference  of  their 

tangents;  that  is,  tan  ^02  =  tan  IF  —  tan  ^0. 

fjN 
Now  it  can  be  easily  shown  that  tan  \0-i.  =  ir-  ;  therefore 

Hi 

gN _    l__gN 
Ri~  2N       E* 


whence 


7?T? 

from  which  R^  =  — '- (259) 

it  —  R\ 

__^  .      „         5730    „        5730    „       5730         ,'  ,      ^ 

Write  ^2  =  -fr-,  Ri  =  -yr-.  R  =  —f^ '  ^^^^  ^"^^^  ^or  B-x. 

Z>2  =  D,  -  A (260) 

in  which  D^.  is  the  degree  of  turnout  from  straight  track. 

Example. — Turnout    from    outside    of    a    4^   curve,    iV  =  8, 
g  =  4.75. 

By  (208").   i?i  =  9.5  X  64  =  608  ft.,  a  9°  2G'  curve. 

By  (260),      7),  =  9°  26'  -  4°  =  5"  26',  for  which  R  =  1054.9. 

By  (258),         Z  =  9.5  X  8  =  76  ft. 

From  (255)  we  have,  by  inverting, 

tan  19  =  jj|-^  =  tan  1°  31' 

auQ,  ?jy  (256), 

I  =  2870  X  sin  1°  31'  =  75.97  ft.. 

a  difference  of  only  0.03  ft.  from  the  value  given  by  (258). 

V 

V 

179.  To  Find  Theoretical  Length  of  Switch-rail  when  the 

Turnout  is  from  a  Curved  Track. 

A  common  tangent  being  drawn  at  the  switch-point,  we  shall 
have,  as  in  169,  for  offset  from  tangent  to  main  curve, 

_  S^ 
^~  2R'' 


FROGS   AND    SWITCHES.  ^  150 


the  offset  from  taugent  to  turnout  is  . 


yi 


When  the  turnout  is  from  concave  side  of  main  line, 


therefore 


whence  S  =  j/  ^J^^^ (261) 

r      It  —  lit 


WritiDg  R=^,R,  =  '^,  and  reducing, 


When  the  turnout  is  from  convex  aide  of  main  line, 

t  =  y^-\'y  =  -^[j^  +  -^y 


whence  ^  =  ^/-^iZ^  ; (263) 

^  R-\-M^ 


from  which 

«="V3i^'«VJ-  •  •  •  '^""^ 

In  (262)  and  (264)  Z>i  is  the  degree  of  turnout  from  straight 
track,  and,  as  these  formuhis  are  identical  with  (216),  it  is  seen  that 
the  theoretical  lengtli  of  switch -rail  on  turnouts  from  curves  is 
the  same  as  on  turnouts  from  straight  line. 

Example.— Find  S  when  t  =  0.42,  ^  =  8,  ^  =  4.75. 

By  (208"),     R,  =  608  feet,  for  which  j9,  =  9°  36'. 


160     A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 
By  (216),  (362),  or  (264), 

S=  107i/-:i?  = 
r    9.43 


22.6  feet. 


180.  Given  the  Distance  p  between  Center  Lines  of  Curved 
Main  Line  and  Side  Track,  the  Frog-angle,  F  {or  Number,  iV^), 
and  Gauge,  g,  to  Find  the  Radius  and  Central  Angle  of  Curve 
beyond  Frog-point, 

First  Case. — Turnout  from  outside  of  main  line. 

In  Fig.  87,  0  is  the  center  of  main  curve,  Oi  the  center  of 


curve  "Whose  radius  is  required.     In  triangle  BOG, 

C0  =  R  +  p-y,     BO:=^R+y, 

By  the  same  reasoning  as  in  177, 

,  ^      2R  +  p^       ,  „         2R  +  P 
cot  ^6=-:: — ^tan^ii^^ 


(265; 


p  -  g         -         2N{p  -  g)' 

In  triangle  OOiB,  OiB  —  R^  —  \g;  then,  by  the  lavs^  of  sines, 

sin  Q 


Ri-k9  = 


(R  +  i9). 


Also, 


sin(i<'+  6) 
BE=2{R-^^g)&m^B 


(266) 
(267) 


FROGS   AND    SWITCHES. 


IGl 


Second  Case. — Turnout  from  inside  of  main  track. 

0,J 


•o 

Fig.  88. 
In  Fig.  88,  we  have  from  triangle  BOE,  reasoning  as  in  178, 

cot  ^9  =  ?^-^li'tan  \F  =  _^-f  ~^.;  .     .     .     (268) 
p  -  g  2Mp  -  g) 

and  from  triangle  OBOi, 

Also,         EC  =  2(i?  -  ig)  sin  ^G, (270) 

and  5^  =  2(i?,  -  ^gr)  sin  4(i^  -  0).  .     .     .     .     (271) 

When  9  is  greater  than  F,  sin  (F—  6)  is  negative,  and  center 
Oi  falls  on  same  side  as  0,  and 


162      A    FIELD-MANUAL   FOR   RAILROAD   ENGINEERS. 


ii^  +  i9  =  ~^^-XR  -  m 


.     .     .     (372) 


sin  {e-Fy 
BE  =  2(i?i  +  y)  sin  i(e  -  F).  .    .    .    (273) 


C.   The  Stub  Lead. 

181.  When  the  frog-numher  exceeds  seven,  the  length  of 
switch-rail  required  to  give  the  necessary  clearance  at  heel  be- 
comes greater  than  is  allowed  in  practice.  To  overcome  this 
ditficulty  slightly  more  curvature  is  given  the  switch-rail ;  more- 
over the  physical  point  of  switch  is  necessarily  some  distance  in 
advance  of  the  theoretical  point.  The  distance  from  heel  of 
switch  to  point  of  main  frog  will  then  be  the  same  as  from  head- 
block  of  stub  switch  to  main -frog  point,  and  is  termed  the  Stub 
Lead.  If  to  this  distance  the  length  of  switch-rail  be  added,  we 
get  the  distance  from  the  head-block  of  a  point  switch  to  the 
point  of  main  frog,  which  is  the  Short  Lead  required  in  practice. 

182.  Given  the  Throw,  i,  the  Gauge,  g,  and  the  Frog-number, 
N,  to  Find  the  Stub  Lead,  s.l. 

In  Fig.  89,  KB  is  the  stub  lead  required;  GIf=  KL,  the  throw. 


Fig.  89. 

From  (207), 

1=  CB  =  2gN, 

and  from  (215), 

s=  CK=2N  Vgi- 

From  the  figure, 

KB=  GB  -  GK, 

or 

8.1.  =  2N{g  -  Vgt) 

(274^ 


FROGS   Al^I)   SWITCHES. 


163 


Formula  (274)  may  be  employed  for  turnouts  from  curves  as  well 
as  straight  lines,  since  it  was  shown  that  the  formulas  from  which 
it  was  derived  may  be  employed  even  when  the  curvature  of  main 
track  is  considerable. 

Below  is  a  table  of  values  of  (g  —  ^y'gt)  for  some  of  the  more 
common  values  of  g  and  t. 

TABLE  OP   VALUES  OF  g  —  |/JT. 


3  Feet  Gauge. 

4  Feet  8i  Inch  Gauge. 

4  Feet  9  Inch  Gauge. 

Throw. 

9  -  Vgt. 

Throw. 

9-  Vgt. 

1 

Throw. 

g  -  Vgt. 

Inches. 
3 

4 

Feet. 
2.13 
2.06 
2.00 

Inches. 
5 

5i 

Feet. 

3.308 
3.239 
3.206 

Inches. 
5 

5| 

Feet. 
3.. 343 
3.275 
3.242 

Example. — Find  the  stub  lead  for  N  =^d>,  g  =  4.75  ft.,  i  =  5 
inches. 

From  the  table,       g  -  \/gl  =  8.343  ft.. 


and,  by  (274). 


s.l.  =  16  X  3.343  =  53.49  ft 


183.  The  Turnout  Table  on  the  next  page  gives  the  frog-angles, 
the  radius  of  center  line  of  turnout  from  a  straight  track  and  its 
degree,  the  theoretical  lead,  the  theoretical  length  of  switch-rail 
for  t  =  5  inches,  and  the  stub  lead  for  certain  values  of  t.  The 
frog-numbers  given  cover  all  the  usual  cases. 

Suppose  it  required  to  find  the  short  lead  for  a  No.  9  frog  and 
5-inch  throw  when  the  gauge  is  4  ft.  9  inches  and  the  length  of 
switch-rail  18  feet.  From  the  table  the  stub  lead  is  60.17  feet; 
hence  the  short  lead  is  60.17  +  18  =  78.17  feet,  as  against  85.50 
ft.  for  the  theoretical  lead. 

Inspection  of  the  table  will  show  that  it  makes  no  very  great 
dilierence  in  the  tabular  quantities  whether  the  gauge  be  taken 
^s  4  feet  8^^  inches  or  4  feet  9  inches.  However,  the  numeiical 
coefficients  in  the  formulas  involving  g  are  somewhat  simpler  for 
the  latter  value. 


164     A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 


TURNOUT  TABLE  FOR  STRAIGHT  TRACK. 

4  FEET  8}4  INCH  GAUGE. 


Degree 
of 

Turn- 
out. 

Theoret- 

Stub-lead for  a 

Throw 

Frog 
No. 

Frog 
Angle. 

Theo- 
retical 
Lead. 

Turn- 
out 
Radius. 

ical 
Switch- 
rail  for 

of 

1 

t  =  5In. 

5  In. 

5]4  In. 

5%  In. 

o 

/ 

feet 

feet 

o             / 

feet 

feet 

feet 

feet 

4 

14 

15 

37.67 

150.7 

38    2 

11.20 

26.46 

25.91 

25.65 

5 

11 

25 

47.08 

235.4 

24  21 

14.01 

33.08 

32.39 

32.06 

5]^ 

10 

23 

51.79 

284.9 

20    7 

15.41 

36.39 

35.63 

35.27 

6" 

9 

32 

56.50 

339.0 

16  54 

16.81 

39.70 

38.87 

38.47 

ei4 

8 

48 

61.21 

397.9 

14  24 

18.21 

43.00 

42.11 

41.68 

1 

8 

10 

65.92 

461.4 

12  25 

19.61 

46.31 

45.35 

44.88 

^14 

38 

70.63 

529.7 

10  49 

21.01 

49.62 

48.59 

48.09 

H" 

1 

9 

75.33 

602.7 

9  301^ 

22.41 

52.93 

51.82 

51.30 

81^ 

6 

44 

80.04 

680.4 

8  25 

23.81 

56.24 

55.06 

54.50 

9 

6 

O-J 

84.75 

762.7 

7  31 

25.21 

59.54 

58.30 

57.71 

9^ 

6 

J) 

89.46 

849.8 

6  45 

26.61 

62.85 

61.54 

60.90 

10 

5 

44 

94.17 

941.7 

6    5 

28.01 

66.16 

64.78 

64.12 

11 

5 

12 

103.58 

1139.4 

5    2 

30.81 

72.78 

71.26 

70.53 

12 

4 

46 

113.00 

1356.0 

4  131^ 

33.61 

79.39 

77.74 

76.94 

13 

4 

24 

122.42 

1591.4 

3  36 

36.42 

86.01 

84.21 

83.36 

14 

4 

5 

131.83 

1845.7 

3    6 

39.22 

92.62 

90.69 

89.77 

15 

3 

49 

141.25 

2118.7 

2  42 

42.02 

99.24 

97.17 

96.18 

4  FEET  9  INCH  GAUGE. 


( 

Degree 

of 
Turn- 
out. 

Theoret- 

Stub-lead for  a 

Throw 

Frog 
No. 

Frog 
Angle. 

Theo- 
retical 
Lead. 

Turn- 
out 
Radius. 

ical 
Switch- 
rail  for 

of 

feet 

<  =  5Iu. 

5  In. 

51^  In. 
feet 

b%  In. 
feet 

o 

/ 

feet 

o         / 

feet 

feet 

4 

14 

15 

38.00 

152.0 

37  42 

11.26 

26.74 

26.20 

25.94 

5 

11 

25 

47.50 

237.5 

24    8 

14.07 

33.43 

32.75 

32.42 

5J^ 

10 

23 

52.25 

287.4 

19  .56 

15.48 

36.77 

36.03 

35.66 

6 

9 

32 

57.00 

342  0 

16  46 

16.88 

40.12 

39.30 

38.90 

6>^ 

8 

48 

61.75 

401.4 

14  16 

18.29 

43.46 

48.58 

42.15 

7 

8 

10 

66.50 

465.5 

12  19 

19.70 

46.80 

45.85 

45.39 

7J^ 

1 

38 

71.25 

.534  4 

10  44 

21.10 

50.15 

49.13 

48.63 

8 

7 

9 

76.00 

608.0 

9  25^ 

22.51 

53.49 

52.40 

51. 8T 

8^ 

6 

44 

80  75 

686.4 

8  21 

23.92 

.56.83 

55.68 

55.11 

9 

6 

22 

>35.50 

769.5 

7  27 

25.32 

60.17 

58.95 

58  36 

9J^ 

6 

2 

90.25 

857.4 

6  41 

26.73 

63.52 

62.23 

61.60 

10 

5 

44 

95.00 

950.0 

6    2 

28.14 

66.86 

65.. 50 

64.84 

11 

5 

12 

104  50 

1149.5 

4  59 

30.95 

73 .  55 

72.05 

71.32 

12 

4 

46 

114.00 

1368.0 

4  11 

33.77 

80.23 

78.60 

77.81 

13 

4 

24 

123.50 

1605.5 

3  34 

36.58 

86.92 

85.15 

84  29 

14 

4 

5 

133.00 

1862.0 

3    4V-0 

39.39 

93.60 

91.70 

90.78 

1  15 

3 

49 

142.50 

2137.5 

2  41 

42.21 

100.29 

98.25 

97.26 

L. 

1 

FROGS   AND   SWITCHES.  165 

184.  To  Stake  Out  a  Turnout. — If  the  positiou  of  bead- block 
is  given,  fix  tlie  frog-point  by  the  foregoiug  table,  remembericg 
that  it  may  be  used  for  turnouts  from  curves,  as  well  as  from 
straight  lines,  without  material  error. 

To  locate  the  rail  between  head-block  and  point  of  switch  it  is 
sufficient  to  do  so  by  offsets  from  main  rail.  Consider  the  equa- 
tion (36)  for  tangent  offsets. 

z  =  \ii^B  for  straight  main  line. 

z  —  \n\I)x  ±  Z>)  for  curved  main  line. 

At  frog-point  z  =  f/,  and  7i  =  /?,  ;  hence 

g  =  ph'A    or     ph^Di  ±  D). 

At  mid-point  of  curve  (practically  mid-point  of  lead),  n  =  \ni , 
and 

2  =  I  .  In.'D,  or  I  .  iWi^(i>a  ±  D)  =  y.  .  (275j 
When  n  —  \ni , 

z  =  l.  j\n,W,  or  I  .  j\7H\D,  ±  D)  =  ^\g.  .  (276) 
When  n  =  f  ?ii , 

z  =  l.j%7H^B,     or    i  .  j\n\D,  ±  JD)  =.  j^,g.    .     (277) 

These  formulas  are  for  the  theoretical  lead,  and  afford  an  easy 
method  of  locating  the  outer  rail  of  turnout  with  all  the  accuracy 
needed  in  practice. 

185.  Curving  Rails. — In  bending  rails  for  curves  the  proper 
curvature  is  determined  by  measuring  the  mid-ordinate  from  a 
cord  held  against  the  inside  face  of  rail-head. 

This  ordinate  may  be  determined  by  (18),  in  which  7i  is  the 
half-length  of  rail  divided  by  100.     For  a  30-ft.  rail, 

M  -  1(0.15)^7)  =  0.0196Z>  =  0.02i)  (nearly).     .     (278) 

From  (209').  i>  =  ^;  and  from  (209"),  ^  =  ^-  Inserting 
either  of  these  values  in  (278)  gives 

J/ =^,,  nearly (279) 


1G6     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

When  the  turnout  is  from  a  curve  compute  M  from  (279),  and 
the  mid-ordinate  for  a  rail  30  ft.  long  on  main  curve  by  (278); 
then  the  mid  ordinate  for  turnout  lail  will  be  the  sum  or  difler- 
ence  of  these  values  according  as  the  turnout  is  from  concave  or 
convex  side  of  main  curve 


\ 


Article  15.  Crossovers. 


186.  To  Locate  a  Crossover  betw^een  Parallel  Straight 
Tracks  w^hen  the  Frog-number,  the  Distance,  p,  between  Cen- 
ters, and  the  Gauge  are  given,  inserting  a  Tangent  between 
Frog-points. 


A 

/ 

i 

•nB 

/ 

'  f 

G 

/f- 

F    / 

'  i 

H> 

/^ 

I 

N 

^v/ 

/    ^>--~^ 

ci 

/ 

M 

K 

^\^^    P 

1 

/ 

Fig.  90. 

In  Fig.  90  it  is  required  to  find  OB  =  KG  =  I,  MK  and  NO, 
also  HK=k. 

In  the  triangle  BPM,  BM=p  —  g\  then 

BE=k  =  BP-EP  1 


or 


k  =  {p  —  g)  cosec  F  —  g  co\.  Fy     .    . 


(280) 


and 


MK=MP-  KP, 


or 


MK  =  (p  —  g)  cot  F—  g  cosec  F. 


(281) 


Jfrom  triangle  OBC  of  Fig.  78, 


^      00      R 

cos  F  =  ^^^  = 


y  _2R-  g 


OB      B+ig      2B  +  g 


(a) 


PROGS   AND   SWITCHES.  167 

In  (a)  write  R  =  2gN^  by  (208),  giving 

4gN^  +  g      4JV*  +  1 

From  Fig  78,  triangle  OBG, 

.    ^      GB  I  21 

Writing  I  =  2gN  and  B  =  2gN^  gives 

From  trigonometry,  taking  the  above  values  of  sin  F  and  cos^, 

Inserting  these  values  in  (280)  and  (281), 

k  =  {p-2g)N+-^ (282) 

MK=ip-2gW-^ (283) 

By  (207),  0B  =  KG=l  =  2gN;  therefore 

IiC=  21  +  MK=4gN+(p-  2g)N -  ^, 

or  NG={p  +  2g)N-^=l-{-p(N  ^  }i\  ,    .    (284) 

Example.— Find  k  and  MK  for  a  No.  8  frog  when  ;>  =  13  ft. 
and^  =  4.75  ft. 

By  (282),  A;  =  3.5  X  8 -r  0.4  =  28.4  feet. 

By  (283).       ifii:  z=  3.5  X  8  -  0.4  =  27.6  feet. 


168    A  field-mani;al  for  railroad  engineers. 

187.  To  Lay  Out  a  Crossover  in  the  Form  of  a  Reversed 
Curve. 

When  j9  is  large,  or  for  otlier  reasons  it  is  desirable  to  get  away 
from  main  track  more  rapidly  than  b}'  the  foregoing  method,  we 
may  lay  out  the  crossover  in  the  form  of  a  reversed  curve. 


A 

A 

0, 

Q 

■0\B         X 

"">^^'"'^ 

1 

1 

M 

/    '  "^ 

I 

1 

1/ 

K 

Fig.  91. 

In  Fig.  91  it  is  required  to  find  QB  =  HE  and  LH. 
Find  OB  =  HE  =1  by  (307),  and  the  radius  00  =  0,C  by 
(208). 
Then,  from  (113),  we  have 

ME  =  2R  sin  a. 


The  angle  a  is  given  by  (112).     Then 

LH  =  2R  sin  a  -  21. 


(285) 


188.  To  Lay  Out  a  Crossover  when  a  Fixed  Length  of  Tan- 
gent must  be  Interposed  between  Points  of  Reversal  of 
Curvature. 

From  the  given  frog-number  determine   the  radius  by  (208) 
then  the  problem  may  be  solved  by  132. 

189,  To  Lay  Out  a  Crossover  in  the  Form  of  a  Reversed 
Curve  -when  the  Tracks  to  be  Joined  are  Ciarved. 

In  Fig.  92  let  the  notation  be  as  shown.  Let  OM  =  B, 
OiM  =  Bi,  O-iP  =  O^C  =  B^. 

0,0^  =  B,  +  B^  =  a, 

00^    =  B-^  p  -  B^  =  b, 


FROGS   AND   SWITCHES. 


109 


00,    =  R-\-  Bx=  C, 

^{a  +  b  -\-  c)  =  5. 


Fig.  92. 
Then,  from  trigonometry, 


cos  hA  =  Y       jc 


cos  ^B 


ac 


(286) 


(287) 


Ml  is  determined  by  178,  and  R^  by  177,  while  R  and  p  are 
given  to  begin  with. 

The  angle  {A  +  B)  determines  the  length  of  arc  CP,  and  angle 
B  the  length  of  arc  MP. 

The  angle  6  is  given  by  (255),  and  0.  by  (245).  Hence  angle 
GOH,  which  determines  the  arc  G'// measured  along  main  track 
between  frog-points,  is 

GOH=  A  -  (9  +  0i). 

The  frog-numbers  at  G  and  B  need  not  be  equal,  only  providing 
that  P  falls  between  G  and  B. 


170     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEP:RS. 


Article  16.     Crossing-frogs  and  Crossing-slips, 

k.    Crossing-frogs. 

190.  When  two  tracks  intersect  each  other  four  crossing-frogs 
are  required  at  the  intersection  of  the  two  sets  of  rails.  The  four 
frogs  are  sometimes  called  a  set  of  crossing-frogs. 

191.  To  Find  the  Length  of  Rails  Intercepted  between  two 

Intersecting  Straight  Tracks  w^hen 
the  Angle  of  Intersection  and  the 
Two  Gauges  are  given. 

In  Fig.  93,  from  triangle  ABH, 

AB=  EG  =  g  cosec  F ;  .     (288) 
and  from  triangle  AEG, 

AE  =  EG  =  g,  cosec  F.      (289) 
Fig.  93. 

192.  Given  the  Angle  of  Intersection,  a,  made  by  the  Center 
Lines  of  a  Straight  and  Curved  Track,  the  Gauges  g\  and  g,  to 
Find  the  Angles  of  the  Set  of  Crossing-frogs. 

In  Fig.  94,  from  the  triangles  OS^Tand  OAH,    '■  .      ' 

(R  +  y)  cos  F=R cos  a  +  ^gi. 


.'.  cos  2^=       „       '  ^^  .    .    (290) 

ii  +  y 


In  like  manner, 

R  cos  a  —  ^gi 


cos  Fi  = 


i?+i^ 


cos  Fa  = 


B-hg 

R  cos  a  -f-  ^gi 


(291) 


_       R  cos  a  —  \gi  ,nr.n^ 

cos  F^  =  — ^-^^.   .     (292) 


(293) 


N  M   HK      11 

Fig.  94. 


From  triangle  BOC  to  find  the  chord  BG. 

BC  =  2(i?  +  Ig)  sin  1{F,  -  F).       ...     (294) 


FROGS   AND    SWITCHES. 


"I  f^"? 


t  L 


Similarly, 

OE  =  2(7?  -  Ig)  sin  ^{F,  -  F^) (295) 

From  triangles  EOM  aud  COL,  we  have 

EG=  ML  =  (R+  y)  sinFr  -  {R-  ^g)  sin  F^.  (296) 
In  like  manner, 

GB  =  JYK  =  {R+  hg)  sin  F  -  {R  -  y)  sin  Fs.      (297) 

193.  Given  the  Angle  of  Intersection,  a,  made  by  the  Center 

Lines  of  Two  Curved  Tracks, 
their  Gauges,  g  and  91  ,  to  Find 
the  Angles  of  the  Crossing-frogs. 

In  Fig.  95,  OA  =  R,  0,A  =  R, , 
and  angle  OAOi  =  a  of  the  triangle 
OAOi  are  given;  whence  OOi  may 
be  determined. 

In  triangle  OBOi  the  side  OB  = 
R  +  y,  0,B=  R,  +  ^gi,  and 
OOi  —  k  are  known,  from  which 
we  can  determine  the  angle  OBOi  = 

Fig.  95.  F- 

In  like  manner  from  the  triangle    OCOi  determine  Fi,  and 
from  triangle  OEOi  find  F^.  F3  may  be  found  from  triangle  OQOi. 

To  find  the  chord  GB  first  find  angle         0. 
BOxO    from  triangle  BOiO,  and  angle 
GOiOfrom  triangle  GOyO;  then 

GB  =  2(i?i  +  ^^0  sin  ^GOiB.  (298) 

In  like  manner, 

EG  =  2(i?i  -  y,)  sin  |JS'0,  G,  (299) 

BG  =  2{R  +  Ig)  sin  ^BOG,      .  (300) 

GE  -  2(i?  -  Ig)  sin  ^GOE.     .  (301) 

When  the  tracks  intersect,  as  in  Fig. 
90,  tlie  solution  is  evidently  similar  to  Fio.  96. 

the  foregoing. 


172    A  field-maxual  for  railroad  engineers. 


B.   Crossing-slips. 

194.  A  Crossing-slip  is  fin  arrangement  of  switch-rails,  in 
connection  with  a  set  of  crossing-frogs,  to  connect  two  tracks 
intersecting  at  a  small  angle. 

195.  Given  the  Angle  of  Intersection  of  Two  Straight 
Tracks,  to  Find  the  Length  and  Radii  of  Curvature  of  Slip-rails. 

In  Fig.  97  determine  EA  and 
AB  by  191 ;  then  assume  GE  or  BH 
(according  as  EA  is  less  or  greater 
tlian  AB)  as  small  as  the  crossing- 
frogs  will  permit.  Draw  the  radii 
HO  and  GO;  AH  =  AG  =  k  h  the 
known  tangent  for  the  central  angle 
F.     Hence 

OG  =  B+y 

=  AH  cot  IF  =2kN,    (302) 


0L  =  R-y  =  2kN -  g. 


(303) 


For  the  theoretical  length  of  rails, 


OH={R+\g)  X  ^1^  X  i^°  =  {R-V  y)  X  ^3.  .    .     (304) 


(305) 


196.  Given  the  Angle  of  Intersection  made  by  the  Center 
Lines  of  a  Straight  and  a  Curved  Track,  to  Find  the  Radii  and 
Length  of  Slip-rails. 

FiKST  Case. — Slip-rails  inside  main  curve. 

In  Fig.  98  determine  the  angles  F  and  Fi  at  B  and  Chj  192. 
Then  assume  KC  as  small  as  constructive  reasons  will  permit. 
Now 


sin^^-OC  =-^ 


EG 


(306) 


b  =  BOK  =  (F,  -F)-  KOC,      .     .     .     (307) 
c  =  KO,H=  F+h  =  F,-  KOC.    .     .    (308) 


FROGS   AND    SWITCHES. 


173 


From  129,  formula  (100), 


B.  +  ^g^O,H=  <-^  +  M"=°^  ^  -  ""'  '\     .     (3091 

^  1  —  COS  c 


Fig.  98. 
For  the  leugths  of  slip-rails, 


HK={R,+  ig)  X 


EL  =  (i?i  -  ig)  X 


57.3 


57.3 


•  • 


(310) 
(311) 


Second  Case. — Slip-rails  outside  main  curve. 

Find  the  angle  F^  at  S,  Fz  at  O,  and  GS  by  the  methods  of 
192.  Assiinie  ^.riV^  as  small  as  constructive  reasons  permit,  and 
calculate  angle  GON,  as  in  (30G)  Then  NOS  =  F^  -  F^  -  GON, 
d=  F^-i-  GON  =^  F^  -  NOS.     By  129,  formula  (106), 


R^^  \g  =  O2M  = 


{R  —  ig)(cos  d  —  cos  F^) 
1  —  cos  d 


(312) 


The  remainder  of  the  solution  is  similar  to  the  first  case. 


197.  Given  the  Angle  of  Intersection  betw^een  the  Center 
Lines  of  Two  Curves,  to  Find  Radii  and  Length  of  Slip-rails. 

First  Case  — Slip-rails  on  concave  side  of  curves. 

In  Fig.  99  take  JJJ  as  small  as  constructive  reasons  permit. 
Join  L  with  0  .  .hen 


174    A  fip:li)-manual  for  kailroad  engineers. 

Determine  angles  COO,,   00,0,  and  side  00,  by  193.     Make 
LM  =  KO,  ;  then 


and 


MOO,  =  LOC+  COOu 


Fig.  99. 

In  triangle  MOO,  two  sides  and  the  included  angle  are  now 
known,  and  tbe  triangle  ma}^  be  solved.  On  is  tbe  center  of 
slip-rail  curves. 


and 


O^MO,  =  0^0, M  =  MOOi  +  M0,0, 
MO^O,  =  180  -  W^MOx. 


From  the  isosceles  triangle  OiO-^M,  in  which  Oi  iff  and  the  three 
angles  arc  known, 

(314) 


MO,  = 


2  sin  IMO^Oi 


Then 


B2  +  l9=  O^L  =  i?i  +  i</  -  MO^, 
E^  -  y  =  0^H=  R,  -  y  -  MO2. 


.     (315) 
.     (313) 


The  central  angle  KO-iL  =  MO^O,  being  known,  Gil  and  KL 
may  be  found  as  in  196. 


FROGS   AND    SWl'KJHKS.  175 

Second  Cask. — ISlip-rails  on  convex  side  of  curves. 

Let  the  dotted  liues  of  Fig.  99  represent  this  case.  Assume 
AQ  aud  compute  angle  AOQ  ;  produce  OQ  to  O2',  the  center  of 
slip-rail  curve  ;  make  O/N  =  O/Oi.  Reasoinng  as  before,  find 
Os'iV  =  0-/0^,  after  which  0^'S,  0^'P,  and  the  lengths  of  TS  aud 
QP  may  be  found  as  in  the  fiist  case. 

Should  the  curves  intersect  as  in  Fig,  96,  no  difficulty  will  be 
found  in  computing  the  radii  and  length  of  slip-rails  by  follow- 
ing the  methods  used  above. 

These  methods  furnish  the  theoretical  length  of  slip-rails  ;  but 
as  the  theoretical  and  ph^'sicai  switch-points  do  not  coincide,  the 
actual  length  will  be  considerably  less, 


CHAPTER  VI. 

CONSTRVCTION. 

Article  17.    Definitions  ;  General  Considerations  ;  Yer- 
TiCAL  Curves  ;  Superelevation  of  Outer  Rail. 

198.  The  work  of  locatiDg  the  center  line  having  been  com- 
pleted, the  field  corps  is  usually  disbanded  and  a  new  one  organ- 
ized. The  Chief  Engineer  still  remains  in  charge,  directing  the 
work  of  construction,  passing  on  bids  and  estimates,  arrangint; 
contracts,  and  attending  to  such  matters  of  importance  as  his  as 
sistants  are  unprepared  or  unauthorized  to  settle. 

199.  A  Division  Engineer  is  placed  in  charge  of  a  considerable 
length  of  line,  made  up  of  several  residencies.  To  him  the  resi 
dent  engineers  make  reports,  and  from  him  receive  directions 
and  orders  relating  to  construction.  These  reports  will  include 
monthly  estimates,  which  are  forwarded  to  the  chief  engineer  for 
inspection  and  approval.  Pay-rolls  for  the  men  employed  are 
made  out  in  the  office  of  the  division  engineer,  and  forwarded  to 
the  chief. 

200.  A  Resident  Engineer  is  placed  in  charge  of  a  few  miles  of 
line,  called  a  Residency,  and  has  direct  charge  of  the  construc- 
tion. He  should  have  at  least  two  assistants— a  rodmau  and  an 
axeman — and  it  will  be  true  economy  to  allow  him  also  an 
assistant  who  can  take  his  place  at  the  instrument  and  assist  in 
superintending  construction. 

The  resident  engineer  is  usually  required  to  set  slope-stakes, 
locate  trestles  and  other  bridges,  tunnels,  culverts,  crossings,  and 
other  features  preceding  track-laying,  and  to  make  all  measure- 
ments upon  which  estimates  are  based  in  determining  the  com- 
pensation of  the  contractor. 

Many  roads  prefer,  especially  on  maintenance  of  w^y,  to  trans- 
pose the  lernis  used  above,  so  thai  the  division  (  ngiueers  report  to 
the  resident  engineer,  whose  residency  may  embrace  several 
divisions. 

176 


CONSTRUCTION".  177 

201.  The  Grade-line  is  determined  from  the  profile,  by  stretch- 
ing a  fine  thread  along  the  paper  and  so  adjusting  its  position 
that  the  proper  relation  between  cut  and  fill  is  obtained,  at  the 
same  time  that  the  maximum  gradient  is  not  exceeded.  The  cuts 
and  fills  should  be  made  as  small  as  possible,  at  the  same  time  that 
badly  broken  or  chopped  grades  are  avoided. 

The  Gradient  is  the  rate  of  change  of  elevation  of  grade-line, 
and  is  usually  expressed  in  per  cents,  a  1.2^  gradient  indicating  a 
rise  or  fall  of  1.2  feet  in  100  feet  horizontal.  When  the  grade  is 
ascending  the  gradient  is  marked  plus,  and  when  descending 
minus. 

The  word  grade  is  frequently  used  instead  of  gradient. 

The  grade-line  should  be  drawn  on  the  profile  in  red  ink,  with 
the  points  of  change  marked  by  a  cross  or  circle,  also  red.  The 
elevations  of  these  points  and  the  gradients  should  be  written  in 
red  above,  or  below,  the  grade  and  surface  lines. 

The  nature  of  the  work  and  the  disposition  of  material  from 
excavations,  and  the  availability  of  outside  material  for  embank- 
ments will  determine  whether  or  not  the  cuts  and  fills  must  balance. 
If  this  would  necessitate  a  long  haul,  it  will  often  be  preferable 
to  waste  material  from  excavation  and  borrow  for  embankments. 

A  Borrow-pit  is  an  excavation,  adjacent  to  the  line,  from  which 
material  is  taken  to  construct  an  embankment.  It  should  be 
separated  from  the  foot  of  the  embankment  by  a  space  termed 
a  Berm,  which  should  increase  with  the  height  of  embankment, 
never  falling  below  a  certain  minimum  width,  say  six  feet. 
Borrow-pits  should  be  regular  in  form,  with  sloping  sides  and 
drained  so  as  to  prevent  water  standing  in  them. 

202.  A  Cross-section  is  a  transverse  section  taken  at  each  sta- 
tion, and  at  intermediate  points  where  the  longitudinal  slope 
changes  considerably,  the  surface  between  adjacent  cross-sections 
being  approximately  such  as  would  be  generated  by  a  straight 
line  moving  on  these  end  sections  as  directors. 

203.  Slope-stakes  are  set  at  stations,  to  mark  the  points  on  cross- 
sections  where  the  side  slope  meets  the  ground-surface.  On 
them  the  cuts  or  fills  are  marked  on  the  inside,  while  the  outsides 
bear  the  station -numbers. 

No  slope-stakes  are  set  at  the  pluses  where  cross-sections  are 
taken,  unless  at  the  top  or  foot  of  bank  where  an  opening  is  left 
for  a  bridge  or  culvert. 


178     A   FIELD-MANUAL    FOR   KAILROAD    ENGINEERS. 

The  Dotes  are  recorded,  however,  in  order  that  the  contents 
m»j  be  correctly  calculated. 

204.  A  Grade-point  is  a  point  on  the  intersection  of  the  plane 
of  the  road-bed  with  the  grouud-surface.  If  the  ground  is  level 
transversely,  a  single  stake  at  the  center,  marked  0.0,  will  suffice 
to  locate  the  point  of  passage  from  cut  to  fill.  When  the  ground 
is  not  level  transversely,  the  line  of  intersection  will  be  oblique  to 
the  axis  of  the  road  and  three  grade-stakes  are  needed,  one  at  the 
center  and  one  at  each  side. 

If  the  width  of  road-bed  in  excavation  differs  from  the  width  in 
embankment,  the  stake  should  be  set  at  the  edge  of  the  widest 
base. 

205.  To  Find  the  Grade-point  when  the  Ground  Slopes 
Uniformly  between  Stations. 


Fig.  100. 

In  Fig.  100  let  AB  be  the  ground-line,  FG  the  grade-line,  and 
E  the  grade-point.  The  horizontal  distance,  y,  from  ^  to  jE'  is 
required.  Let  the  cut  at  J.  be  hi  ,  the  fill  at  B,  h-i ,  and  the  length 
of  prismoid  I.  Draw  BG  parallel  to  CF.  From  the  similar  tri- 
angles ^^i^  and  ABG 


X  = 


Ih, 


hi  -\-  hi' 


(317) 


If  the  ground  does  not  slope  uniformly,  the  point  E  must  be 
found  by  trial,  such  that  the  rod-reading  equals  the  difference 
between  height  of  instrument  and  elevation  of  grade. 

206.  Vertical  Curves. — The  angle  formed  by  the  junction  of 
two  grade-lines  should  be  rounded  off  either  by  substituting 
several  small  chuuges  for  the  one  large  one,  or,  preferably,  by  in- 


CONSTRUCTION, 


179 


scrting  a  regular  curve.  Where  the  algebraic  difference  of  gradi- 
ents is  less  thau  0.3,'^  uo  curve  will  be  needed,  while  for  larger 
differences  the  length  of  vertical  curve  should  vary  with  that 
difference,  unless  the  circumstances  of  the  case — such  as  the 
proximity  of  other  vertical  curves,  or  a  bridge — should  prescribe 
its  length.  In  any  case  the  length  may  be  either  assumed,  or  a 
given  rate  of  change  per  station  fixed  upon  and  the  length  com- 
puted. 

The  parabola  is  especially  well  adapted  for  vertical  curves,  be- 
cause of  the  ease  with  which  any  correction  may  be  found  when 
one  is  known,  since,  as  will  presently  be  shown,  the  corrections 
vary  as  the  square  of  the  distance  from  the  point  of  tangency. 
A  second  property  of  this  cuive  enables  us  readily  to  find  the 
correction  at  the  vertex,  or  meeting-point  of  grade-lines.. 


Fig.  101 

In  Fig.  101  let  ^Cand  CBhe  the  intersecting  grade-lines,  and 
AFB  the  curve  substituted  for  them.  Produce  AC  to  ^  to 
meet  a  vertical  through  B.  Draw  the  vertical  CG.  Then  will 
CF  =  FG  =  w  by  the  second  property  referred  to.  Since 
measurements  are  made  horizontal!}'',  the  similar  figures  AGG 
and  AEB  furnish  the  relation  CF  =  iGG  =iEB.  Calling  the 
algebraic  difference  of  gradients  d,  and  the  length  of  curve  21, 


7/1  =  -Id. 
4 


(a) 


If  the  rate  of  change  of  gradient  per  station  be  a,  it  is  evident 
that 


The  equation  of  the  parabola  referred  \o  A  as  origin  may  be 
written 


(c) 


180     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

To  fiud  the  correction  IIK  --  z  !it  ;i  distance  .i"  from  A,  we  have 
from  the  similar  triangles  AHL  and  ACG, 

HL  =  CG^  =  2m~ {d) 

C  6 

But  KL  =  y,  and  z  =  HL  —  KL,  or,  inserting  values, 

_  2771X  __  /2m.r       m.i'^\ 


2 


or  2  =  m- (318) 

Insert  the  value  of  d  from  (h)  in  {a)^  and  the  resultant  value  of 
m  in  (318);  then 


fa 

x^ 

a 

z 

— 

2 

X 

r 

— 

-%^ 

(318') 


'\* 


When  x  —  \  station,  2,  =  \a  :  when  a;  =  2  stations,  22  =  3rt,  etc. 

It  will  only  be  necessary  to  figure  corrections  for  one-half  the 
curve,  as  they  are  the  same  for  corresponding  points  each  side  of 
the  vertex.  If  preferred,  however,  all  corrections  maybe  com- 
puted from  the  first  tangent  produced. 

Example. — A  +  0.9^  meets  a  —  0.6,^  grade  at  sta.  181^  the  ele- 
vation of  which  is  91.0  ft.  liequired  the  corrections,  and  cor- 
rected grade  elevations  for  points  100  ft.  apart. 

Here  the  algebraic  difference  of  gradients  isO.9  — (—  0.6)  —  l.o. 
Suppose  a  be  taken  as  0.25,  or  the  lengtlif  of  curve  as  6  stations. 

Formula  (r/,)  gives  ??i  =  ^X3xl-5  =  l.f25  feet. 

At  the  P.C,  sta.  178,  s  =  0;  at  179,  (318)  or  (318')  gives  2,  = 
0.125;  at  180,  2,  =  4  X  0.125  =  .50.  The  original  and  corrected 
grade  elevations  are  as  follows: 

Sta.  178         179  180  181  182  18?,  I8t 


Original  elevation.     88.3       89.2 

Corrections 0.0         0.125 

Corrected  elevat'u    88.30     89.075 


*  If  a  circle  be  taken  as  the  joininp:  curve  we  niaj'  derive  (318')  bj'  finding 
E  in  terms  of  a,  then  writing  />  =  5730  -^  R,  and  n  =  a:,  in  formula  (86). 


90.1 

91.0 

90.4 

89.8 

89  2 

0.50 

1.125 

0.50 

0.125 

0  0 

89.60 

89  875 

89.90 

89.675 

89  2C 

CONSTRUCTION. 


181 


Example   2. — A   +  0.3^    meets  a  +  1.1^   grade  at    sta.    312, 
whose  elevation  is  155  0      Fiiul  correctious. 


Take  a  =  0.2  in  this  case;  Iheu  I  = 


+  0-3-(+l.l)_o 


2  X  .2 
lions.     The  corrected  grade  heights,  etc.,  will  be  as  follows 


=  2  sta- 


sia. 


310 


311 


312 


313 


314 


154.7 

155.0 

156.1 

157.2 

0.1 

0.4 

0.1 

0.0 

154.8 

155.4 

156.2 

157.2 

Original  elevation ...       154.4 

Corrections 0.0 

Corrected  elevation 154.4 


The  reason  for  adding  the  corrections  in  this  case  will  be  evi- 
dent from  a  figure. 

The  table  below  gives  the  corrections  in  feet,  for  certain  alge- 
braic  differences  of  gradients  and  lengths  of  curve,  at  intervals  of 
50  ft.  each  way  from  the  vertex.  When  the  difference  of 
gradients  is  plus,  tlie  correction  must  be  suhtracted  from  the 
original  grade  elevation  ;  when  llie  difference  is  minus,  the  cor- 


TABLE    OP   CORRECTIONS    FOR   VERTICAL   CURVES. 


1  ... 

-i^ 

Algebraic 
Diflferenc 
of    Grad 

eiits. 

Kate  of 
Change  o 
Grade  pe 
Station. 

Distance  from  Vertex  in 

Feet. 

0 

50 

100 

150 
0.01 

200 

250 

300 

350 

400 

0.3 

0.075 

0.15 

0.08 

0.04 

0 

0  4 

.10 

.20 

.11 

.05 

.01 

0 

0.5 

.125 

.25 

.14 

.06 

.02 

0 

0.6 

.15 

.30 

.17 

.08 

.02 

0 

0.7 

.175 

.35 

.20 

.09 

.0;! 

0 

0.8 

.20 

.40 

.23 

.10 

.03 

0 

0  9 

.225 

.45 

.25 

.11 

.03 

0 

1.0 

.25 

.50 

.28 

.13 

.03 

0 

11 

.1833 

.83 

.57 

.37 

.21 

.09 

.02 

0 

1.2 

.20 

.90 

.63 

.40 

.23 

.10 

.03 

0 

1.3 

.2167 

.98 

.68 

.44 

.24 

.11 

.03 

0 

1.4 

.2333 

1.05 

.73 

.47 

.26 

.12 

.03 

0 

1.5 

.25 

1.13 

.78 

.50 

.28 

.13 

.03 

0 

1.6 

.2667 

1 .20 

.83 

.53 

.30 

.13 

.03 

0 

1.7 

.2833 

1.28 

.89 

.57 

.32 

.14 

.04 

0 

1.8 

.30 

1.35 

.94 

.60 

.34 

.15 

.04 

0 

1.9 

.2375 

1.90 

1.46 

1.07 

.74 

.48 

.27 

.12 

.03 

0 

2  0 

.25 

2.00 

1..53 

1.13 

.78 

.50 

.28 

.13 

.03 

0 

2.1 

.2625 

2.10 

1.61 

1.18 

.82 

.53 

.30 

.13 

.03 

0 

2.2 

.275 

2.20 

1.68 

1.24 

.86 

..55 

.31 

.14 

.03 

0 

2.3 

.2875 

2,. 30 

1.76 

1  29 

.90 

.58 

.32 

.14 

.04 

0 

2.4 

.30 

2.40 

1.S4 

1.35 

.94 

.60 

.34 

.15 

.04 

0 

2.5 

.3125 

2.50 

1.91 

1.41 

.97 

.63 

.35 

.16 

.04 

0 

2.G 

.325 

2.60 

1 

1.99 

1.46 

1.02 

.65 

.37 

.16 

.04 

0 

182     A   FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

lection  must  be  added.  Similar  tables  for  other  lengths  of  curve 
or  differences  of  gradients  maybe  computed  by  the  eugineer,  aud 
time  in  the  field  saved  by  their  use.  In  setting  grade-stakes,  it 
will  be  well  to  set  them  50  ft.  apart  on  vertical  curves,  though  to 
allow  for  the  vertical  curve  at  each  regular  station  will  suffice 
when  cross-sectioning. 

207.  Elevation  of  Outer  Rail  on  Curves. — In  138  it  was 
sliown  that  the  superelevation  of  outer  over  inner  rail  might, 
for  standard  gauge  track,  be  given,  nearly  enough,  by  the  formula 


e  = 


dR' 


(134) 


in  which  e  is  the  elevation  in  feet,  and  Fthe  velocity  in  miles  per 

5730  * 

hour.     Writing  E  =  ——-  in  this  formula  gives 

e  =  0. 000058  F'i> C319) 

The  following  table  has  been  computed  by  formula  (319). 


TABLE  OF  SUPERELEVATIONS  OF  OUTER  RAIL. 


r 
V'm 

Degree  of  Curve. 

Miles 
per 

Hour. 

1" 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

12° 

20 

.02 

.05 

.07 

.09 

.12 

.14 

.16 

.19 

.21 

.23 

.28 

30 

.05 

.10 

.16 

.21 

.26 

.31 

.37 

.42 

.47 

.52 

.63 

40 

.09 

.19 

.28 

.37 

.46 

.56 

.65 

.74 

.84 

.93 

50 

.15 

.29 

.44 

.58 

.73 

.87 

1.02 

1.16 

CO 

.'21 

.42 

.63 

.84 

1.04 

1.25 

Since  grade-stakes  are  set  at  the  edge  of  base  it  will  be  neces- 
sary to  determine  the  difference  between  these  elevations  and  the 
elevations  of  center  line.  Calling  this  difference  h,  the  half-base 
h,  and  the  distance  between  centers  of  rail-heads  for  standard 
gauge  4.9  feet,  we  shall  have,  from  similar  triangles. 


\  =  -T-r  e  =  0.2be  (nearly). 
4.9  \         J/ 


(320) 


If  the  inner  rail  is  required  to  remain  at  grade  (320)  will  become 

7i  =  0.2be  ±  0.5e, (320') 

according  as  the  grade-stake  is  to  be  set  at  outer  or  inner  edge  of 
base. 


CONSTRUCTION".  183 

Example, — What  will  be  the  value  of  7i  when  e  =  .46,  the 
base  being  14  feet? 

By  (320),  A  =  0.2  X  7  X  0.46  =  0.64  feet.  The  outside  is  this 
much  higher  than  the  center,  the  inside  edge  this  much  lower. 

The  superelevation  of  outer  rail  should  be  computed  for  the 
highest  speed  at  which  trains  are  to  be  run  over  the  curve;  the 
maximum  allowed  in  practice  rarely  exceeds  8  inches,  since  a 
greater  elevation  would  endanger  the  slow-running  freight  trains. 
Even  when  the  theoretical  superelevation  is  given  tiie  outer  rail, 
it  is  more  worn  than  the  inner  one,  either  because  there  are  other 
forces  acting,  or  because  of  the  sliding  action  of  the  outer  wheel 
due  to  imperfect  adjustment  where  the  original  coning  has  been 
destroyed  by  wear. 

Engineers  sometimes  elevate  the  outer  rail  1  inch  per  degree  up 
to  3°,  and  make  6  =  3i  inches  for  a  4°  curve,  4  inches  for  a  5° 
curve,  and  4|  inches  for  a  6°  curve.     Still  other  rules  are  in  use. 

If  transition-curves  are  not  employed,  the  difference  of  eleva- 
tion is  the  same  from  P.O.  to  P.T.,  fading  out  to  nothing  on 
tangent.  The  elevation  begins  on  tangent  from  50  to  200  feet 
back  of  P.O.,  depending  on  the  amount  the  outer  rail  is  to  be 
raised. 

208.  Easing  Grades  on  Curves. — To  compensate  for  the 
increased  resistance  due  to  curvature,  it  is  customary  to  reduce 
the  grade  on  curves.  This  resistance  is  taken  to  vary  directly  as 
the  curvature;  a  rule  often  used  is  to  reduce  the  gradient  0.05 
foot  per  degree  of  curve 

\  Article  18.    Earthwork. 

A.  Setting  Slope-stakes 

209.  Slope-stakes  are  set  at  the  points  where  the  side  slopes 
meet  the  ground-surface,  to  mark  the  limits  of  the  excavation  or 
embankment,  and  to  show  the  constructor  what  the  cut  or  fill 
must  be.  In  Fig.  102,  if^i^  represents  the  ground-surface,  IIBG 
the    grade-surface.     Let   AB  =  h   be    the    center   height.     Let 

HL 

=  s  be  the  side  slope,  which  varies  with  the  nature  of  the 

KL 

material;  for  earth-excavation  the  side  slope  will  average  about  1 

to  1,  so  that  s  =  1,  while  for  ordinary  earth-embankment  it  will 


184     A    FIELD-MANUAL   FOR   RAILROAD    ENGINEERS. 

average  about  1^  to  1,  so  that  s  =  1^.  The  side  height  for  level 
sections  is  the  same  as  the  center  and  may  be  found  for  any 
section,  so  that  the  distance  LH  is  required.     From  the  equation 


j*"^-*^ 6---*|<---6^--->t<--As--»4 

i  '  i  '  i 


I 


Fig.  102. 


defining  s,  HL  =  KLs=Tis.     Let  the  base  EC  =  2b;    the  "dis- 
tance out  "  from  center  is 

d.o  =  BL  =  h  -\-  hs. 

210.  Surface  Inclined. — Where  the  ground  slopes  transversely 
the  position  ot  the  slope-stake  cannot  be  found  from  the  center 
height  unless  the  slope  of  the  ground-surface,  us  well  as  the  side 
slope,  is  known.  The  slope-stakes  can  be  most  easily  and 
rapidly  set  by  trial. 


Fig.  103. 

In  Fig.  103,  FAE  is  the  ground-surface,  AB  =  7i  the  fill  at 
the  center.  We  have  to  find  the  distances  out  of  ^and  F  from 
A,  and  the  side  heights  ME=  hi  ,  and  NF  =  li^.  Let  OP  repre- 
sent the  plane  of  the  instrument  at  a  height  U.I.  above  the 
datum,  obtained  from  the  known  elevation  of  a  bench  or  turn- 
ing-point. PB  is  the  height  of  the  i)]ane  of  the  instrument  above 
the  grade.     Call  PB  the  Station  Constant  (s.c). 


COXSTRUCTION".  185 

The  fill  at  A  will  evideut^y  equal  the  rod  rcadiui;  less  the 
slatiou  constant.     Mark  this  ou  the  center  stake. 

Since  the  ground  slopes  downward  from  J.  to  E,  the  distance 
out  will  be  greater  than  for  a  level  section,  while  for  F,  on  the 
higher  side,  it  will  be  less. 

Suppose  we  take  a  reading  ^^at  a  distance  out  —  h-\-lis\  the 
fill  at  that  point  is  LK  —  QK  —  QL  =  r  —  s.c,  and  the  corre- 
sponding distance  out  is  d.o.  =  b  -{-  KL  X  s,  which  is  greater 
than  All,  since  LK  is  greater  than  h.  If  now  a  reading  is  taken 
at  the  distance  out  6  +  *  X  LK,  we  shall  have  a  fill  greater 
than  LK,  luiless  the  ground  is  level  from  ^to  E,  and  therefore 
b  -\-  s  X  LK,  the  distance  out  actually  used,  will  be  less  than 
that  called  for  by  the  reading.  However  we  shall  have  obtained 
a  closer  approximation  to  the  position  of  E,  and  by  repetitions  of 
this  process  may  come  as  close  to  its  true  position  as  the  con- 
ditions require. 

The  same  thing  can  be  accomplished  more  rapidly  by  estimat- 
ing the  fall,  by  the  eye,  from  ^  to  a  point  b  -f  hiS  out,  then  mul- 
tiply this  estimated  fall  by  the  slope  and  add  to  b  -\-  hs.  Take  a 
reading  r  at  this  distance  out;  then  compute  the  d.o.  for  the  fill 
?•  —  s.c.  and  note  if  this  agrees  with  the  actual  d.o.  If  it  does 
not,  make  a  new  trial  with  this  reading  as  a  guide. 

For  ordinary  work  the  actual  and  computed  distance  out  should 
be  such  that  if  the  rod  were  held  at  the  computed  distance  the 
new  distance  would  not  differ  more  than  a  tenth  from  that  just 
computed.  The  stake  is  then  set  at  the  computed  distance  out. 
After  a  little  practice  it  will  be  found  that  the  second  setting  of 
the  rod  may  usually  be  made  to  fall  as  close  to  the  true  position 
as  the  limit  requires. 

When  the  stake  is  marked  and  driven  the  cut  or  fill  at  that 
point  and  the  distance  out  are  recorded  in  the  notes. 

As  an  example  suppose  26  ■=  14  feet,  *  =  H  (i.e.,  slope  1^-  to  1), 
H.I.  =  187.3,  grade  elevation  =  184.0.  The  station  constant  is 
s.c.  =  187.3  —  184.0  =  3.3.  Suppose  the  rod  at  center  to  read 
8.5  ;  the  fill  will  be  8;5  —  3.3  =  5.2,  which  mark  on  stake  as 
"  F.  5.2. "  The  distance  out,  if  section  were  level,  would  be  d.o.  — 
7 -f  5.2  X  1.5=  14.8;  but  suppose  the  ground  rises  and  we  esti- 
mate the  rise  as  1  foot,  which  multiplied  by  s  gives  1.5  feet  to  be 
subtracted  from  14.8,  since  this  is  on  the  higher  side  of  center 
for  a  section  in  embankment.  Let  the  reading  at  13  3  out  be 
7.7,  which  gives  a  fill  of  7.7  —  3.3  =  4.4  feet,  calling  for  a  d.o.  = 
7-1-4.4  X  1.5  =  13.6.      This  shows  we  are  too  far  in.  but  as  a 


18G     A    FIELD-MAX  UAL    FOR    RAILROAD    EXGINEERS. 


reading  further  out  will  be  less,  giving  a  correspoudingly 
smaller  do.,  we  try  a  reading  at  13.5  feet  out.  Suppose  the  read- 
ingto  be  7.6;  the  fill  will  be  7.6  —  3.3  =  4.3,  calling  for  a  distance 
out  of  13.45  feet,  which  agrees  almost  exactly  with  the  trial  dis- 
tance. The  stake  is  marked  "  F.  4,3,"  and  the  result  recorded 
in  the  cross-section  book. 

On  the  other  side  of  the  section  suppose  we  estimate  the  fall  to 
be  1.5  feet  in  15;  we  should  try  a  reading  at  13.8  +  1.5  X  1.5  =  16.1, 
say  16.0  feet.  Let  this  reading  be  9.0  ;  the  fill  will  be  9.0  -  3.3 
=  5.7  feet,  calling  for  a  do.  =  7  +  5.7  X  1.5  =  15.6,  which  shows 
our  readinir  was  taken  too  far  out.  Trv  a  reading  at  15.4,  which 
suppose  8.9  ;  the  fill  is  8.9  —  3.3  =  5.6,  and  the  d.o.  =7+5.6 
X  1.5  =  15.  4,  which  agrees  exactly  with  the  trial  distance. 

In  excavation  the  method  of  proceeding  is  the  same  as  in  em- 
bankment, except  that  s  has  generally  a  different  value.  For  solid 
rock  s  is  usually  I,  that  is,  the  slope  is  taken  as  ^  to  1;  for  loose 
rock,  gravel,  and  ordinary  earth  the  slope  may  be  taken  as  1  to  1. 

The  station  constant  in  cuts  is  always  positive,  and  the  rod 
reading  has  to  be  subtracted  from  it  to  obtain  the  out.  In  fills, 
when  the  II. I.  is  greater  than  the  grade  height,  the  fill  equals  the 
difference  of  the  rod  reading  and  the  station  constant.  When 
the///,  {'^less  than  the  grade  height  the  rod  reading  plus  the 
s.c.  gives  the  fill. 


211.  The  Notes  may  be  kept  in  the  form  below,  which  repre- 
sents one  page  of  the  cross-section  book.  The  cut  or  fill  is  written 
above  the  line,  the  distance  out  below,  A  plus  sign  indicates  a 
cut,  a  minus  sign  a  fill 


Sta. 


IGl 

162 

-f  20 

-f  48 

+  66 

163 


Ground. 

Grade. 

178.8 

184.0 

181.6 

183.0 

182.2 « 

O 

»^ 

182.5  ' 

183.5 

185.0 

182.0 

Left. 


-  4.4 


13.6 

-  0.8 

8.2 

0.0 

9.0 

-f  0.9 

9.9 

_L  2.4 

11.4 

4-4.4  -f  2.8 

13.4       10.0 

Center. 


-  5.2 

-  1.4 

-  0.6 
0.0 

-f  1.2 
+  3.0 


Right. 


5.6 


15.4 

10.0 

-  1.0 
8.5 

-  0.4 
7.6 
0.0 
9.0 

+  4.3   +2.6 


6.1       11.6 


CONSTRUCTIO]Sr.  187 

212.  Irregular  Sections. — Wbeu  readings  are  taken  only  at 
the  center  and  sides  it  is  termed  a  "three-level  section."  Very- 
irregular  ground  may  require  several  more  readings  in  order  to 
determine  its  area  ;  in  this  case  a  reading  is  taken  at  each  change 
of  surface  in  the  section,  and  the  cut  or  fill,  together  with  the  dis- 
tance out,  recorded— the  distance  being  measured  from  the  center 
to  the  point  where  the  rod  was  held  in  taking  the  reading. 

When  the  base  cuts  the  ground-surface  the  section  is  partly  in 
excavation  and  partly  in  embankment,  but  each  side  will  be 
staked  out  in  the  manner  described  above.  The  distance  of 
grade-point  from  center  must  be  found  and  recorded. 

213.  Staking  Out  Openings. — Where  openings  are  to  be  left 
for  trestles,  culverts,  and  other  structures,  stakes  must  be  set  to 
mark  the  limits  of  the  embankment.  Stakes  marked  T.  B.  are 
set  at  the  center  and  sides  to  fix  the  place  where  the  top  of  bank 
is  to  end  ;  other  stakes,  marked  F.  S.,  are  set  at  the  foot  of  slope, 
the  plus  at  which  they  fall— together  with  the  distance  out  from 
center— being  recorded  in  the  note-book.  The  slope  of  the  toe 
of  dump  should  be  the  same  as  the  side  slope. 

214.  Marking  Stakes.  — All  slope  and  toe  stakes  that  limit 
excavation  or  embankment  should  be  driven  with  tops  inclined 
outward  from  the  center.  The  cut  or  fill  is  marked  on  inside  in 
plain  figures  preceded  by  the  letter  C.  or  F.  as  being  more 
easily  understood  b}^  the  contractor  than  the  plus  and  minus 
signs  used  in  the  notes.  The  reverse  side  should  bear  the  station 
number. 

215.  Shrinkage — Gi  owth.  — It  must  be  remembered  that  earth- 
work  in  embankment  will  settle,  or  shrink  in  volume,  even  after 
having  been  compacted  by  the  feet  of  the  teams  during  construc- 
tion. Where  the  fill  is  not  great,  allowance  may  be  made  for 
shrinkage  when  setting  grade-stakes,  but  in  heavy  fills  allowance 
should  be  made  when  the  stakes  are  set  for  construction.  The 
proper  allowance  will  vary  with  the  nature  of  the  material,  but 
about  10  per  cent  will  be  a  fair  average.  The  contract  should 
always  specif}'^  the  amount  of  shrinkage  to  be  allowed  on  par- 
ticulai  works.  If  the  earth  is  measured  in  the  borrow-pits,  an 
equivalent  allowance  shoukl  be  made,  since  earth  is  more  com- 
pact in  embankment  than  l)efore  excavating. 

With  rock,   however,  it  is  found  that  the  volume   increases 


188     A    FIELD-MANUAL   FOR   RAILROAD    EXGIXEERS. 

after  excavation,  and  this  increase  is  termed  growth.  The  size  of 
the  fragments  will  determine  the  growth,  wbicli  will  vary  from 
one  half  to  five  eighths  of  the  original  volume — the  larger  the 
fragments  the  greater  the  increase.  Little  or  no  allowance  need 
be  made  for  settlement  when  placed  in  embankment. 

216.  Borrow-pits  should  be  regular  in  form,  particularl}'  if 
the  volume  of  earth  moved  is  to  be  measured  in  the  borrow-pit. 
They  should  be  properly  drained  to  prevent  water  standing  in 
them  and  should  have  an  ample  berm  between  edge  of  pit  and 
foot  of  slope,  the  width  of  berm  increasing  with  the  height  of 
embankment. 

B.  Areas  of  Sections. 

217.  Before  the  volume  of  earth  in  excavation  or  embank- 
ment can  be  computed  the  area  of  each  cross-section  must  be 
found.  To  do  this  divide  the  section  into  triangles  and  trape- 
zoids, find  the  area  of  each  separately,  and  take  the  sum.  To 
shorten  the  calculations  a  few  simple  rules  will  be  deduced. 

When  the  center  and  side  heights  are  equal  we  have  a  one- 
level  section;  when  the  center  and  side  heights  differ  it  is  a 
three-level  section;  where  the  height  is  found  at  five  places  in 
the  section  it  is  a  five-level  section ;  and  so  on. 

218.  To  Find  the  Area  of  a  Three-level  Section. 


Fig.  104. 


In  Fig.  104  the  area  ABCDF  is  required.  Draw  EA  and  EG, 
dividing  the  area  into  four  triangles.  "With  the  notation  of  the 
figure  it  is  seen  that  triangles  (1)  and  (2)  have  equal  bases  h  and 


CONSTKCCTION. 


ISO 


altitudes  //i  and  hi  ,  while  (3)  and  (4)  have  the  common  base  Zio 
and  altitudes  Bi  =  b  -\-  IIi  s  and  di  =  b  -\-yiiS.     By  geometry, 


Area  {l)+(2)  =  b 


III  +  hr 

2     '' 


Area  (3)  +  (4)  =  /^o ^~  =  ho  g      — -- 

The  area  of  the  whole  section  is  therefore 

A  =  ,?i+h  +  nD^^  ^  ^11^  ^  ;JM:(^+^.  (32,) 
-»  *  2  2 

This  formula  affords  an  easy  method  of  obtaining  the  area  of  a 
three-level  section,  aud  when  written  in  words  becomes  the  follow- 
ing 

llvM,Y..  — Multiply  the  half -sum  of  the  side  heights  by  the  half -base 
and  to  this  add  the  product  of  the  center  height  by  the  half -sum  of 
the  distances  out;  the  result  icill  be  the  area. 

If  the  linear  measurements  are  in  feet,  the  area  will  be  in 
square  feet. 

When  Hy  =  hi  =  ho ,  then  Di  =  d^  =  d  =  b+  hos  and  (321)  re- 
iuces  to 

A  =  bho  +  ^lod  =  ho{b  +  d)  =  ho{2b  +  7ioS).     .     .     (322) 

The  section  is  now  a  trapezoid  or  one  level  section  for  which 
'322)  may  be  deduced  by  the  usual  rule  for  the  area  of  a  trapezoid. 

219.  Area  of  Five-level  Section. 

Fig.  105  represents  this  case,  where  we  may  evidently  divide 


N^ M 


D- — ^-rf-i-^^    •• 


Fig.  105. 


he  area  into  triangles  and  trapezoids,  computing  the  area  of 
?ach  separately  and  taking  their  sum  for  the  whole  area.  The 
'oliowing  simpler  method  may  be  preferred: 


190     A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


Write  the  notes  as  in  field-book,  except  that  center  height  is 

placed  over  zero  and  an  additional  -  is  written  at  each  end  as 

below. 

L  C  R 


Beginning  at  the  center,  multiply  heights  by  distances  out  in 
pairs  as  indicated  by  the  sloping  lines,  the  products  of  members 
connected  by  full  lines  being  plus  and  of  those  connected  by 
dotted  lines  minus.  Half  the  sum  of  the  products  will  be  the 
area,  thus: 


^  =  2 


(323) 


The  grouping  is  symmetrical  for  areas  each  side  of  center,  as 
(323)  exhibits;  so  it  will  be  sufficient  to  show  that  the  rule  is  cor- 
rect for  either  side.  Divide  the  figure  into  trapezoids  and  tri- 
angles as  shown;  then 

Area  trapezoid  BCKM  =  |(7io  +  Hi)Di  , 

ABM]^  =  MR,  +  E,){D^  -  Di\ 
"     triangle  ALN       =  ^H^{D2  -  b). 

The  two  trapezoids  include  the  triangle  ;  hence  the  latter  must 
be  subtracted  from  their  sum.  Doing  this  and  simplifying,  we 
have 

Al  =  W^oD,  +  E^D^  -  DxH2  +  S^b), 

which  is  the  same  as  results  in  (323). 

220.  General  Formula  for  Areas. — The  method  of  219  may 
be  applied  to  any  section  no  matter  how  irregular.  Suppose 
there  have  been  n  levels  taken  on  one  side  exclusive  of  the  center 
height;  the  notes  would  appear  as  below: 


0 


dy 


hi 

57 


h„. 


dn-l 


Tin 
dn 


0_ 

h 


Expanding  in  the  same  manner  as  in  219, 

'^R=l\!Jlo^i+TLxdi+      .  .  hn-idn-^hnb)  —  {dJl'i-\- 


.  dn-xhn)l  (324) 


To  show  that  (324)  gives  the  true  area,  consider  that  we  have  n 


CONSTRUCTION.  191 

trapezoids  whose  area  is  pc^sitive,  aud  oue  triangle  whose  area  is 
negative  and  equal  to  J  lin{dn  —  b). 
Writing  out  the  area,  we  have 

Ar  =  4[(7/o  -f  Ai)<Z.  +  (A.  -f  h^){ih  -  (U) 

+  .  .   .  (/i„_i  +  hn){dn  —  dn-l)  —  Kidn  -  b)]. 

Performing  the  indicated  operations  aud  simplifying, 

Ar  =  \[{hodi-\-h^d.,  +  .  .  .  +  hn-idn-\-  hnh)  —  {dji2  .  .  .  +  dn-\hn), 

which  is  the  same  result  obtained  in  (324). 

Evidently  n  may  have  any  positive  integral  value. 

If  preferred,  the  cross-sections  may  be  plotted  on  cross-section 
paper  and  the  area  read  off  by  means  of  a  planimeter. 

221.  Tables    of  Areas    of  Level  Sections,  and   the   Three- 
level    Correction. — Formula  (332)  may   be   employed    in   com 
putiug  the  areas  of  level  sections  for  any  values  of  b  and  s. 

Table  XVII  gives  the  areas  for  a  few  of  these  values.  When 
many  sections  are  to  be  figured  it  will  be  well  for  the  engineer  to 
compute  the  necessary  tables,  provided  he  is  unable  to  secure 
published  ones  for  the  particular  bases  and  slopes  he  is  working 
with.  It  is  not  witljin  the  scope  of  this  volume  to  give  the 
variety  of  tables  needed;  they  are  published  elsewhere. 

The  area  of  three-level  sections  may  be  found  from  the  areas  of 
level  sections  by  the  aid  of  a  suitable  correction.  Let  the  height 
used  in  entering  the  tables  of  level  sections  be  the  mean  height  of 

the  three-level   section,  /t„i  = -^ ^" -';  the  corresponding 

area,  by  (322),  is 

A'  =  ?im{2b  -f  7ims)  =  2b7i„i  +  7im^s (a) 

The  true  area  is  given  by  (321): 

4  4 

=  2b7im  +  27iq7i„,s  —  7io'^s (b) 

From  equations  (a)  and  (b)  the  correction  is 

c  =  A'  -  A  =  {7hn'  -  2AoAm  H-  7l^)s  =  {7i,n  -  7i,ys.    (325) 


193     A    FIELD-MAXUAL    FOR    RAILROAD    ENGINEERS. 

Table  XYIII  was  computed  by  (325),  aud  gives  values  of  c 
wbicli  are  always  positive,  and  whicb  must  be  subtracted  from 
the  tabular  area,  found  by  entering  the  table  with  the  mean 
height  of  section,  in  order  to  get  the  true  area. 

Example.— The  side  heights  for  a  till  having  a  14  ft.  base  and 
side  slopes  1*  to  1  are  8.6  aud  16.4  feet,  while  the  center  height  is 
7.7  ft.     The  mean  height  is 

8.6  +  2x7.7  +  16.4 

hm  = —    =  10.1  ft., 

4 

for  which  Table  XVII  gives  A'  =  294.4  sq.  ft.  For  the  correc- 
tion, 7im  -  7^0  =  10.1  -  7.7  =  2.4  ft.,  for  which  Table  XVIII 
gives  1^  =  8.6  sq.  ft. 

The  true  area  is  now  A  =  294.4  —  8.6  =  285.8  square  feet. 


C.   Volume  of  Earthwork. 

222.  Cross-sections  must  be  taken  at  all  full  stations  and  at  in- 
termediate points,  or  pluses,  where  tliere  is  a  change  in  longitudi- 
nal slope.  It  will  be  well  in  any  event  to  take  them  so  close 
together  that  the  difference  in  end  heights  .should  not  exceed 
about  live  feet.  The  time  consumed  in  making  these  intermediate 
measurements  will  be  more  than  offset  by  the  reliability  of  the 
results.  A  few  of  the  more  usual  methods  of  estimating  quan- 
tities will  be  given  here. 

223.  Averaging  End  Areas. — This  is  the  easiest  of  application 
aud  therefore  the  most  generally  used,  but  is  open  to  the  objec- 
tion that  it  gives  inaccurate  results'.  However,  when  bids  are 
based  upon  it,  both  parties  to  the  contract  agreeing,  it  would 
seem  to  answer  as  well  as  any  other  method. 

If  Ai  and  J.2  are  the  end  areas  aud  I  the  length,  we  shall  have 

y^A^+A^^ (326) 

Stated  in  words,  (326)  yields  the  following 

Rule. — MuUiply  the  half -sum  of  the  end  areas  by  the  axial 
length  of  jirismoid  ;  the  resrilt  will  be  the  volume. 

If  areas  are  in  square  feet  and  length  in  feet,  the  volume  will 
be  in  cubic  feet  ;  to  reduce  1o  cubic  yards  divide  by  27. 


CONSTRUCTION.  193 

224.  Prismoidal  Formula. — The  parallel  scetious  should  be 
so  takeu  that  tlie  surface  bounding  the  volume  to  l)e  measured 
may  be  supposed  to  be  generated  by  a  straight  line  moving  on 
the  bounding  lines  of  llie  sections  as  directors  in  such  a  manner 
as  to  return  to  its  original  position.  Such  a  figure  is  called  a 
•prismoid. 

Tlie  heiffht  of  anv  section  intermediate  between  the  end 
sections  is  a  function  of  its  distance  from  either  end,  and  the 
area  of  that  section  will  be  a  quadratic  function  of  its  distance 
from  either  end. 

Now  we  know  from  mechanics  that  Simpson's  (Newton's)  Rule 
will  hold  for  any  function  not  higher  than  the  third  ;  so  for  the 
mean  area  this  rule  yields 

Mean  area  = , 

where  A^  and  A^  are  the  end  areas,  Am  the  middle  area ;  hence 
we  have  for  the  volume 

F  =  (^1  +  4^m  +  ^2)|,        .    .     .     (337) 

in  which  I  is  the  axial  length  of  prismoid. 

The  same  result  may  be  obtained  geometrically  by  dividing  the 
prismoid  into  prisms,  wedges,  and  pyramids,  and  applying  the 
usual  rule  for  volumes. 

Let  the  end  areas  be  ai,  a^,  fti',  ^2',  etc.,  and  the  mid-areas 

^m,  (fm  >  etc. 

For  the  prism  ai  =  a-i  =  am ;  hence  the  volume  is 

V  =  aj  =  (ui  -f  4a,H  +  a-i)~x (a) 

For  the  wedge  ay'  —  2am'  and  a-i'  =  0;  tlierefore 

v' =  a,' J  ={((,'-{- 4:am' -i- a^')-^.      ...     (6) 

For  the  pyramids  a/'  =  4a»i"  and  a^"  =  0;  hence 

^'  =  a/'l  ==  («/'  +  4a„r  +  ci,")^.       .     .     (c) 


194     A    FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 

Adding  {a),  (b),  and  (c),  the  total  volume  is 

V  =v-i-v'  +  v" 

=  [(«i+«i'+«i")+4(rt,„-ha,«'+«,„")+(«2+a2'+a2")]-|     .    («f 
But  «i  +  tti'  +  «i"  =  -4i 

flm  -|-  <lm    ~r  ^^"t      -—   Am, 

and  tta  +  «2'  +  "2"  =  --12  ; 

therefore  V=  {Ai  +  44^  +  ^2)  •  -^, 

o 

the  same  as  (337). 

Stated  in  words  there  results  the  following 

Rule. — To  the  sum  of  the  end  areas  add  four  times  the  mid-area, 
multiply  by  the  Lengthy  and  divide  by  6.  The  result  will  be  the 
volume. 

To  reduce  to  cubic  yards,  divide  by  27. 

Formula  (327)  contains  three  terms,  the  middle  area  being 
derived  from  the  cross-section  notes  for  the  end  sections  at  the 
expense  of  some  little  trouble.  In  the  attempt  to  simplify  this 
formula  Dr.  George  Bruce  Halsted  in  1881  published  a  two-term 
prismoidal  formula,  giving  the  volume  in  terms  of  one  base  and 
a  section  at  two  thirds  of  the  length  of  the  prismoid,  the  formula 
being 

F=(^, +  3.4§)|=(3^t  +  ^2)|   .     .     .     (328) 

In  1894  Professor  AV.  H.  Echols  showed  by  the  aid  of  higher 
mathematics  that  an  indefinite  number  of  two-term  formulae 
might  be  derived.  The  same  results  were  established  in  1895  by 
Professor  T.  U.  Taylor  by  elementary  mathematics. 

None  of  these  two-term  formulae  have  so  far  been  placed  in  a 
form  suitable  for  application  to  earthwork  measurement,  owing 
to  the  difficulty  of  finding  the  area  of  the  auxiliary  section. 

In  fact  the  only  objection  to  the  use  of  (327)  is  the  loss  of  time 
required  in  obtaining  the  mid-area  and  the  uncertainty  as  to  its 
accuracy  in  the  case  of  very  irregular  sections. 

For  three-level  ground  we  may  construct  a  section  having 
heights  that  are  means  between  corresponding  end  heights,  but 
for  very  irregular  sections  there  may  be  uncertainty  as  to  w'rat 
heights  must  be  averaged  to  obtain  the  mid  section  heights.  For 
any  other  than  the  mid-sections  the  heights  are  obtained  with 
more  difficulty. 


CONSTRUCTION. 


195 


225.  Form  of  Notes. — The  record  of  areas  and  volumes  may 
be  kept  iu  the  form  below,  which  represents  the  cross-section 
book.witli  the  necessary  columns  added. 


Sta. 

Ground 

Grade. 

91 

188.5 

182.0 

92 

185.4 

181.0 

93 

180.0 

180.0 

94 

173.0 

179.0 

L. 


+6.3 
15.2 

±iii 
13.1 

+2.0 
11.0 
+  1^ 
10.0 
0.0 

9  0 
-2  1 
10.2 
-4  2 


13.3 


c. 

R. 

End 
areas. 

Mid- 
areas. 

Exc. 
cu.yds. 

+6.5 

+8.9 
17.9 

+  175.53 

+5.3 

+7.5 
16.5 

+130.64 

+4.1 

+6.1 
15.1 

+89.96 

486.4 

+2.1 

+3.0 
12.0 

+41.10 

0.0 

0.0 
9.0 

0.0 

158.8 

-3.0 

-6.0 

-4.3 

13.5 

-8.6 

19.9 

-144.40 

-57.95 

Emb. 
cu.yds. 


232.2 


If  the  method  of  averaging  end  areas  is  employed,  the  column 
of  mid-areas  will  not  be  needed,  and  may  even  be  omitted  when 
computing  by  the  prismoidal  formula.  In  this  case  the  notes  for 
mid-section  and  the  mid-area  should  be  written  in  red  ink. 

An  office  record  should  be  kept  iu  addition  to  the  record  in  the 
cross-section  book,  to  which  it  will  not  be  necessary  to  transfer 
the  elevations  of  ground  and  grade.  If  preferred,  the  areas  and 
volumes  may  be  kept  ouiy  iu  the  office  record,  omitting  them  in 
the  cross-section  book. 


226.  Prismoidal  Correction.— The  time  and  labor  required  to 
obtain  the  area  of  the  mid-section  makes  the  use  of  the  prismoidal 
formula  objectionable;  for  this  reason  the  method  of  averaging 
end  areas  is  most  often  employed.  The  difference  in  the  two 
methods  will  not  be  great,  provided  the  difference  in  end  heights 
is  not  over  3  or  4  ft. ;  it  should  never  exceed  5  ft. 

When  the  difference  exceeds  this  a  considerable  error  is  intro- 
duced by  the  use  of  (326).  It  will  generally  be  sufficient  to 
average  end  areas  and  then  apply  a  correction  if  the  result  must 
be  free  from  large  errors. 

(a)  Correction  for  Level  Sections.— Between  two  level  end 
sections  the  volume  is  made  up  of  one  prism,  one  wedge,  and  two 
pyramids.     For  the  prism  and  wedge  the  true  volume  is  given  by 


196     A    FIELD-AIANUAL    FOR    RAILROAD    ENGINEERS. 


averagiug  end  areas,  but  for  the  pyramids  the  error  is  easily 
shown  to  be 

2{Ho  -  ho){H^  -  7io)s  (  I        I 


2 


"o  —  q"  1  cubic  feet. 


or 


0 

G  =  {Ho  — hnY  stt—-^  cubic  yards.     .     .     .    (329) 


The  table  below  gives  the  correction  C  in  cubic  yards,  com- 
puted by  (329),  for  a  few  values  of  Hq  —  lir^ ,  when  s  =  1  and  I  = 

Is 
100  ,  for  any  other  length  and  slope  multiply  by  r^. 

TABLE   OF   PRI8MOIDAL   CORRECTIONS   FOR   LEVEL   SECTIONS. 


Ho  —  ho. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

1 

0.6 

0.7 

0.9 

1.0 

1.2 

1.4 

1.6 

1.8 

2.0 

2.2 

1^ 

2.5 

2.7 

3  0 

3.3 

3.6 

3.9 

4.2 

4.5 

4  8 

5.2 

3 

5.6 

5  9 

6.3 

6.7 

7.1 

7  6 

8.0 

8  5 

8.9 

9.4 

4 

9.9 

10.4 

10.9 

11.4 

12.0 

12.5 

13.1 

13.6 

14.2 

14.8 

5 

15.4 

16.1 

16  7 

17.3 

18.0 

18.7 

19.4 

20.1 

20.8 

21.5 

6 

22.2 

23.0 

23.7 

24.5 

25.3 

26.1 

26.9 

27.7 

28.5 

29.4 

7 

30.2 

31.1 

32.0 

32.9 

33.8 

34.7 

35.7 

36.6 

37.6 

38  5 

8 

39.5 

40.5 

41.5 

42.5 

43.6 

44.6 

45.7 

46.7 

47.8 

48.9 

9 

50.0 

51.1 

52.2 

53  4 

54  5 

55.7 

56.9 

58.1 

59.3 

60.5 

"10 

61.7 

63.0 

64.2 

65  5 

66.8 

68.1 

69.4 

70.7 

72.0 

73.3 

11 

74.7 

76.1 

77.4 

78.8 

80.2 

81  6 

83.1 

84.5 

86.0 

87.4 

12 

88. 9 

90.4 

91.9 

93.4 

91.9 

96.5 

98.0 

99.6 

101.1 

102  7 

13 

104.3 

105.9 

107.6 

109  2 

110.8 

112.5 

114.2 

115  9 

117.6 

119.3 

14 

121.0 

122  7 

124.5 

126.2 

128.0 

129.8 

131.6 

133  4 

135.2 

137.0 

15 

138.9 

140.7 

142.6 

144.5 

146.4 

148.3 

150.2 

152.2 

154.1 

156.1 

(b)  Correction  for  Three-level  Sections. — Formula  (329)  or 
the  foregoing  table  may  be  used  in  determining  the  correction 
for  three-level  sections  when  these  sections  are  somewhat  similar 
and  the  corresponding  heights  not  very  different.  A  general 
formula  may,  however,  be  derived. 

Let  the  center  and  side  heights  at  one  section  be  Ho,  Hi.  and 
//a,  respectively,  and  let  the  distance  between  slope-stakes  be 
W  =  Di  -\-  D-2;  let  the  corresponding  heights  at  the  other  end 
section  be  h^,  Jii,  and  hi  with  a  distance  between  slope-stakes  of 
w  =  clx  -\-  d-i. 

By  formula  (321)  the  areas  will  be: 


h  W 

Area  at  first  end  =  -{Hi  +  H^)  +  -^H^,  . 


(«) 


CONSTRUCTION. 


197 


Area  at  secoud  end  =  -^(A,  -f  //.)  -| — ho,     .     .     .(^) 


4  X  mid  area  =  4 


V       2       "^       2 


=  b{IL  +  Ai  +  //.  +  h,)  ^{W^w) ^^^".    {c) 
From  (a),  {h),  aud  {c)  the  volume  by  the  prismoidal  lule  is 


V  = 


Piih + h,  +//.+A.)+  ^y[ih+^fj+Mo+'^\ 


Xg.  (^0 


From  {a)  aud  {b),  by  averaging  end  areas, 


-(//,  +  7i.  +  ^a  +  7i2)  +  -—  +  — - 


I 

X3 


=  [f 6(7/i  +  h,  +  i?2  +  7^2)  +  i  TF^o  +  |«^/^o]  X  g.     .     .    («) 
From  ((?)  and  (e)  the  correction  is 

F'  -  F  = 


-?(i7„  -  h,)  +  |(7i„  -  ^„) 


I 


=  (i7o  -  7io)(  W  -  iO)  X  r^  cu.  ft. 
The  correction  in  cubic  yards  is 


(330) 


I 


when  I  -  100, 


(7=(^„-7a^-^)X  j^^^;    .     .     (331) 


C  =  0.31(/7o  -  K){  W  -  w) (331') 


This  correction  may  be  either  positive  or  negative. 
ExA-MPLE. — Compute   the  correction   for   the   two    prismoids 
below 

Sta.  L. 

+  5.4 


160. 
161. 
162. 


14.4 

+  10.8 
19.8 

+  5.2 
14.2 


C. 

+  3.0 


R. 
+  6.6 


+  7.0 
+  9.0 


15.6 

+  11.2 
20.2 

+  4.8 
13.8 


19S     A    FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

For  tlie  prismoid  between  160  and  161 

100 
C  -  (3.0  -  T.OaSO.O  -  40.0)  X  z-^ ^  -  —  I'-^-S  cu.  yds. 

For  the  prismoid  between  161  and  162 

C  =  (7.0  -  9.0)(40.0  -  28.0)  X  ttt^^  =  -  7.4  cu.  yds. 

When  the  correction  C  is  positive  it  must  be  subtracted  from, 
and  when  negative  added  to,  the  volume  as  found  by  averaging 
end  areas;  this  is  evident  from  (330).  Cwill  be  negative  when 
the  smaller  center  height  and  greater  width  between  slope-stakes 
occur  at  the  same  station,  as  is  illustrated  in  the  example. 

The  general  tendency,  however,  with  the  method  of  averaging 
end  areas  is  to  give  volumes  that  are  too  large,  and  the  error  in- 
creases with  the  square  of  the  difference  in  end  heights,  as  is  evi- 
dent from  (329). 

227.  "When  the  center  and  sides  of  roadbed  do  not  pass  from 
cut  to  till  at  the  same  station  we  have  a  volume,  part  excavation, 
part  embankment,  between  the  same  pair  of  sections,  such  as  is 
illusti-ated  in  Fig.  106. 

Between  sections  ABC  and   ELF  ML  the  solid  havin?  bases 


Fig.  106. 

ABC  and  FEB  is   embankment,  while  the  pyramid  FML-A  is 
in  excavati»)n. 

For  such  cases  as  lliis  the  method  of  averaging  end  areas  is 
most  in  error,  particularly  if  the  ground  have  a  sliarp  longitudinal 


COXSTELCTION. 


199 


as  well  as  Irausverse  slope      Whatever  method  is  employed,  the 
excavations  and  embankments  must  be  separately  computed. 

228.  Tables  of  Volumes  for  Level  Sections  and  Equal  End 
Areas  may  be  \ised  iu  making  iiieliminary  estimates.     The  aver- 
age center  height  for  one  or  moie  stations  is  taken  from  the  pro 
file  and  the  volume  at  once  read  off  from  tables,  such  as  Table 
XIX. 

Table  XX  may  be  used  in  finding  the  volume,  after  having 
averaged  the  end  areas,  and  a  correction  made  by  226  if  desired. 

229.  Side  Ditches  iu  cuts  have  a  constant  cross-section,  and 
hence  a  constant  volume  for  each  full  station.  Their  contents 
are  separately  computed  and  added  after  the  other  computations 
have  been  made.    They  need  not  be  shown  iu  cross-section  notes. 

230.  Earthwork  on  Curves.  —  In  computing  quantities  on 
;urves  the  end  sections  are  assumed  to  be  parallel,  and  the  axial 
distance  between  sections  taken  as  the  length  of  the  prismoid. 
If  the  volume  be  taken  as  generated  by  a  moving  section,  and  the 
center  of  gravity  of  this  .'section  lie  always  on  a  vertical  line  passing 
through  the  axis,  this  method  gives  correct  results  ;  otherwise  not. 
The  result  will  be  too  small  or  too  large  according  as  the  center  of 
gravity  falls  without  or  within  the  center  line  of  curve. 

If  the  volumes  are  computed  by  averaging  end  areas,  it  will  be 
A  useless  refinement  to  apply  a  curvature  correction  ;  but  if  the 
prismoidal  formula  is  employed,  and  accuracy  is  desired,  it 
should  be  applied,  especially  if  the  work  be  in  rock. 


Fio.  107. 

To  find  the  curvature  correction  (c.c.)  consider  Fig.  107,  which 
represents  the  ynean  section  of  the  prismoid. 


200     A   FIELD-MANUAL    FOR    RAILROAD    ENGINEERS. 

Tlie  portion  ABIIEG  has  its  ceuter  of  gravity  on  the  Hue  BF 
{^Z?^  having  the  same  slope  as  BA) ;  hence  the  path  of  its  ceuter 
of  gravity  will  be  the  same  length  as  the  axis  of  the  prisnioid, 
and  there  will  be  no  error  in  the  computed  volume  generated  by 
this  portion.  In  the  triangle  i>C// draw  BK  to  the  midpoint  of 
GH.  The  center  of  gravity-  of  this  triangle  is  at  M,  two  thirds  of 
the  distance  BK  from  B.  Now,  by  Guldin's  rule  (theorem  of 
Pappus)  the  volume  generated  equals  the  area  multiplied  by  the 
path  of  the  center  of  gravity,  the  center  of  rotation  being  in  the 
plane  of  the  area. 

Draw  BL  horizontal  and  take  N  on  a  vertical  through  M;  let 
the  angle  in  degrees  at  the  center  be  5. 

The  volume  generated  by  the  triangle  BCH  is 

V=BCHf^iR+BN). 

But  the  calculated  volume  is 

Fo  =  BCE  X  I  =  BCh'^R. 


Hence  the  curvature  correction  will  be 

180 


C.C.  =   V-    r,  =  BCII^r::,BN. 


%  2   d,-\-d^      d,  +  d. 

But  BN=  ^BL  =  ^^.  -^  =  —^ 

•  ••  «•<'•  =  ~BCH{d,  +  d.iYf  =  .OOQBCB{d^  +  d.,)B°.    (332) 
o40 

When  the  sections  are  100  ft.  apart  6  =  I)  and  the  correction 
becomes 

c.c.  =  .OOQBCH{di  +  d.)D (332') 

The  area  of  the  triangle  BCH  is  easily  seen  to  be 

•      A  =  l[b{?i.2  -  h,)  +  7>.o{d-2  -  (h)]-     .     .     (333) 

If  the  triangle  BCH  is  on  the  convex  side  of  curve  the  correc- 
tion must  be  added,  if  on  the  concave  side  it  must  be  subtracted. 

For  light  work  the  correction  is  small,  but  for  heavy  work  with 
steep  transverse  slope  on  sharp  curves  it  ma}'  be  considerable. 
In  practice  we  may  use  the  middle  for  the  mean  area  without 
material  error 

Example. —Find  the  correction  per  station  on  an  8"  curve,  28 


il 


CONSTRUCTION 


201 


ft.  base,  side  slopes  1|-  to  1,  iuside  height  10  ft,,  outside  height 
30  ft.,  end  sections  equal. 

231.  Overhaul. — Coutract  prices  are  usually  based  on  a  certain 
maximum  length  of  haul,  and  all  material  carried  farther  than 
this  is  termed  overhaul,  for  which  the  contractor  receives  extra 
compensation. 


Fig.  108 

In  Fig.  108  let  ABhe  the  length  of  free  haul,  the  points  A  and 
B  being  fixed  on  the  profile  so  that  the  volume  ACE  equals  the 
volume  CBK\  this  may  be  done  by  trial  computations,  or  closely 
enough  iu  some  cases  from  the  profile  alone.  Let  the  mass 
EFIIA  be  removed  to  BKLM,  and  let  the  centers  of  mass  in  the 
two  positions  be  at  g  and  g^  respectively;  the  length  of  overhaul 
to  be  paid  for  will  be  GG,  -  AB  =  GA  +  BG^.  To  find  g  and 
gx  accurately  requires  that  the  sum  of  the  moments  of  the  ele- 
mentary masses  equal  the  moment  of  the  whole  mass  with  respect 
to  any  chosen  point.  It  will  answer  in  practice  to  multiply  the 
volume  per  station  by  the  distance  of  its  center  of  mass  (found 
by  dividing  the  station  length  in  the  inverse  ratio  of  its  end  areas) 
from  some  selected  point,  as  G,  and  equate  this  to  the  product  of 
the  whole  mass  by  the  unknown  distance  of  its  center  of  gravity 
from  the  same  point,  then  solve  for  this  distance.  Indeed,  it  will 
answer  iu  most  cases  to  find  a  point  that  divides  the  mass  into 
two  equal  parts  and  treat  this  as  the  center  of  gravity;  such  a 
point  may  be  readily  found  by  trial. 

Sometimes  it  is  specified  that  the  overhaul  must  be  found  by 
finding  the  distance  of  the  center  of  gravity  of  the  wJiole  mass 
moved  from  the  center  of  gravity  of  the  same  mass  after  deposit- 
ing in  embankment  and  deducting  from  this  the  length  of  free 
haul,  the  remainder  being  called  the  overhaul.  This  is  the  easier 
method,  but  requires  every  yard  moved  to  be  carried  the  entire 
lengtli  of  free  haul  before  any  overhaul  whatever  is  counted. 

KxAMiM.E  -  Let   AEVn  =.  5000  cu.  yds.,   GA  =  200  ft.,   G,B 


202     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 

=  300  ft.,  and  the  price  paid  for  overhaul  1  cent  per  cubic  yard 
per  100  ft. 

The  additional  compensation  above  the  contract  price  will  be 
200  +  300 


100 


X  5000  X  .01  =  $250.00. 


Article  19.  Grade  and  Ballast  Stakes,  Culverts, 
Bridges,  and  Tunnels. 

232.  Grade  and  Center  Stakes. — After  the  excavations  and 
embankments  have  been  brought  approximately  to  the  level 
called  for  on  the  cross-section  stakes,  the  engineer  must  go 
carefully  over  the  road,  setting  center  stakes  every  hundred  feet 
on  tangents  and  flat  curves,  and  every  oC  or  25  feet  on  sharp 
curves — the  distance  between  center  stakes  depending  on  the 
sharpness  of  the  curves.  On  tangents  it  will  be  sufficient  to  drive 
a  grade-stake  beside  each  center  stake,  so  that  its  top  will  be  at 
tbe  height  to  which  the  finished  surface  must  come,  due  allow- 
ance being  made  for  shrinkage. 

On  curves  grade-stakes  must  be  set  at  each  side  a  distance 
equal  to  the  half-base  from  the  center;  the  proper  elevation  or 
depression  of  these  stakes  must  be  found  by  207,  formula  (320). 

The  P.C.'s  and  P.T.'s  are  recovered  by  means  of  the  reference- 
points  set  during  location. 

233.  Ballast-stakes  are  set  on  the  completed  sub-grade  at  the 
proper  width  of  ballast-base — just  as  in  slope-staking — with  their 
lops  at  the  level  of  the  final  grade.  They  should  be  set  at  inter- 
vals of  50  ft.  ou  tangents  and  flat  curves,  and  at  25  ft.  on  sharp 
curves. 

234.  Track  Centers  are  set  for  the  guidance  of  trackmen  as 
soon  as  the  road-bed  is  ready  to  receive  the  cross-ties  and  rails. 

235.  The  Opening  left  for  a  culvert,  drain,  or  trestle  bridge  is 
measured  from  top  of  bank  to  top  of  bank;  the  manner  in  which 
it  should  be  staked  out  is  described  in  213. 

A  note  of  the  size  of  drain  and  the  material  of  which  it  is  to  be 
built,  whether  glazed  earthenware  pipe,  box  drain,  stone  culvert, 
etc.,  should  be  made  in  the  note-book  opposite  the  notes  for  the 
opening. 

After  the  culvert  or  drain  has  been  built  the  earth  is  filled  in 


CONSTRUCTION.  203 

over  and  around  it,  and  face  or  wing  walls  built  to  protect  the 
bank  at  the  points  where  culvert  or  drain  meets  its  face. 

For  trestle  bridges  it  must  be  remembered  that  the  bank-sills 
set  back  from  the  top  of  bank  a  distance  sufficient  to  give  firm 
bearing,  usually  about  6  ft,  for  ordinary  earth,  and  allowance 
made  therefor  in  staking  out  the  opening.  The  length  of  open- 
ing is  designated  by  the  number  of  bents  between  bank-sills; 
thus  a  12-bent  opening,  where  the  distance  between  bents  is  14 
ft.,  would  be  13  X  14  —  12  =  170  ft.  The  bent  spacing  depends 
upon  the  size  of  timbers  available  and  upon  the  weight  of  loco- 
motives to  be  run  over  the  road. 

Whatever  the  nature  of  the  structure,  ample  waterway  should 
dvvays  be  provided  for  the  heaviest  storms;  failure  to  do  this  is 
;,he  cause  of  many  a  costly  wreck.  , 

Center  stakes  are  set  for  each  trestle-bent,  and  if  piles  are  to  be 
driven  a  stake  should  mark  the  position  of  each  pile.  If  the 
bridge  is  not  at  right  angles  to  the  stream  it  will  often  be  best  to 
Bet  the  bents  askew,  but  this  should  be  avoided  whenever  possible. 
After  the  piles  have  been  driven  cut-off  levels  are  given  by  the 
engineer,  for  which  a  tack  is  set  in  the  pile  at  a  definite  distance 
below  the  point  of  cut-off,  allowance  being  made  for  cap, 
stringer,  etc.  If  the  bridge  is  on  a  grade,  the  rate  of  rise  per  bent 
must  be  figured  out  and  allowed  for.  On  curves  the  proper 
superelevation  of  outer  rail  must  be  computed  by  the  method  of 
207. 

For  details  of  trestles  see  Foster's  Trestle  Bridges. 

236.  The  Piers  and  Abutments  for  truss  bridges  must  be  very 
accurately  located,  the  spacing  being  done  with  a  steel  tape 
whose  constants  are  known,  and  the  center  and  limits  being 
marked  by  stakes.  On  tangents  the  centers  are  easily  located 
and  referenced,  but  on  curves  this  is  not  so  easy,  as  the  center  of 
track  cannot  be  taken  as  the  center  of  pier  on  account  of  the 
clearance  necessary  for  trains. 

Bridges  on  curves  should  be  avoided  whenever  possible,  but 
when  they  cannot  be  avoided  the  centers  of  piers  are  to  be  placed 
at  the  intersection  of  pier-axis  and  "bridge-chord." 

In  Fig.  109  ABC  is  the  center  line  of  track,  AE  and  CF  the 
pier-axes.  At  the  mid-point  of  the  arc  AG  the  tangent  EF, 
parallel  to  AC,  is  drawn ;  make  AN  =  NE  =  CL  =  LF,  and  draw 
NL,  which  is  the  bridge-choid.  The  points  N  and  L  are  the 
centers  of  the  piers. 


204     A    FIELD-MANUAL   FOR   RAILROAD    EXGIXEERS. 


Should  L  or  N  be  inaccessible,  the}'"  may  be  locp.ted  from  a 
point  P  on  some  accessible  portion  of  the  curve.  To  do  this  lake 
PQ  perpendicular  to  LN,  such  that 

PQ  =  RB- KB  =  E{veTsb-iYeTsa);   .     .     (334) 
then  will 

QL  =  QK  +  KL  =  Rlsin  h -\- sin  a).     .     .     .     (335) 

The  manner  of  building  the  piers,  determining  the  nature  of 
the  foundation,  and  erecting  the  bridge  come  properly  within 


\ 


s 


\  ,y'    \if^^'    I 


I       / 


y 


/ 


•4/ 


Fio.  109. 

the  province  of  the  bridge  engineer  and  require  too  much  space 
to  be  outlined  here.  For  preliminary  estimates  it  will  often  be 
suflScieut  for  the  locating  engineer  to  make  soundings  with  gas- 
pipe  in  order  to  determine  the  depth  to  a  suitable  foundation 
and  the  nature  of  the  overlying  deposits,  the  core  forced  up 
within  the  pipe  serving  for  the  latter  purpose. 

237.  Tunnels,  like  bridges,  require  great  nicety  in  the  meas- 
urements by  which  they  are  constructed.  The  angular  measure- 
ments should  be  made  with  the  best  available  transit  in  the  best 
possible  adjustment,  and  repetitions  and  reversals  made  to  elimi- 
nate errors  as  much  as  possible.  Linear  measurements  should  be 
made  with  a  steel  tape  the  constants  of  which  are  known,  so  that 
correction  may  be  made  for  temperature,  etc. 

If  headinETS  are  to  be  driven  from  the  ends  and  an  unobstructed 
view  of  the  summit  is  obtainable,  a  point  may  be  fixed  in  the 
same  vertical  plane  as  the  axis  of  the  road,  and  will  serve  for 
giving  the  alignments  of  the  headings. 

Sometimes  several  points  must  be  located  on  the  mountain  in 
the  plane  of  the  axis,  and  triaugulation  resorted  to  to  secure  the 
desired  end. 


CONSTRUCTION.  205 

The  most  accurate  work  can  often  be  doue  at  night,  sightings 
being  made  to  a  pliimmct-laivip,  or  in  the  early  morning  before 
the  sun's  heat  has  produced  great  changes  in  the  density  of  the 
air. 

Within  the  tunnel,  alignment  is  made  by  sighting  to  a  plummet- 
lamp  suspended  from  a  plug  V't  into  the  roof.  Work  is  usually 
carried  on  from  both  ends,  so  that  it  is  necessary  to  secure  accurate 
alignment.  If  the  entire  tunnel  is  on  a  tangent,  this  is  not 
ditlicult  when  working  from  the  ends:  but  when  the  tunnel  is  on 
a  curve  (the  curve  falling  most  often  at  the  ends),  or  when  align- 
ment must  be  transferred  down  a  shaft,  the  operation  is  much 
more  difficult. 

The  Mont  Cenis  Tunnel,  over  seven  miles  long,  was  constructed 
from  the  ends — one  end  being  on  a  curve — yet  there  was  no 
trouble  in  making  a  fit  where  the  headings  met. 

When  headings  are  driven  from  the  foot  of  a  shaft  it  is 
necessary  to  secure  a  point  in  the  surface  on  each  side  of  shaft 
in  the  plane  of  tunnel-axis  and  to  transfer  these  points  by  plumb- 
lines  to  the  bottom.  By  connecting  these  transferred  points  the 
direction  of  the  axis  may  be  secured.  In  the  Hoosac  Tunnel  a 
line  was  transferred  down  a  shaft  1000  ft.  deep  and  carried  2050 
ft.  with  a  final  error  of  only  nine  sixteenths  of  an  inch. 

Levels  are  run  over  the  surface  with  great  care,  and  may  be 
transferred  down  a  shaft  by  measuring  its  depth  with  a  rod  or 
steel  tape.  The  grade  of  the  bottom  must  be  sufficient  for 
drainage. 

The  dimensions  of  the  tunnel  will  depend  on  the  height  of 
engines  and  the  purposes  for  which  it  was  intended. 

For  detailed  information  regarding  tunnels  the  student  is 
referred  to  Drinker's  and  Sims's  books  on  the  subject  and  to  the 
current  engineering  journals. 

Article  20.  ^Fontiily  and  Final  Estimates. 

238.  Monthly  Estimates  are  made  by  the  engineers  in  charge 
of  construction  about  the  end  of  each  monlh,  and  upon  these  the 
division  engineer  bases  his  estimate,  which  he  forwards  to  the 
chief  engineer  for  approval.  The  contractor  receives  his  compen- 
sation some  days  later,  usually  about  the  loth  or  20th  of  the 
month  following.  Monthly  estimates  should  always  be  based  on 
actual  measurements  and  never  guessed  at,  jiarticularly  if  several 
classifications  are  to  be  made.     The  total  quantity  of  work  done 


206     A    FIELD-MA NCAL   FOR    RAILROAD    ENGINEERS. 

or  material  delivered  is  to  be  estimated,  then  the  difference 
between  any  estimate  and  the  last  preceding  one  will  be  the 
estimate  upon  which  the  contractor  receives  his  installments. 

239.  For  Earthwork,  measurements  (when  needed)  are  only 
approximate,  but  it  is  best  to  make  them  with  level  and  tape 
even  for  monthly  estimates.  It  will  be  sufficient  to  compute 
volumes  by  averaging  end  areas,  no  attention  being  paid  to  the 
prismoidal  correction.  Care  must  be  taken,  however,  that  such 
estimates  are  not  in  excess — in  fact,  it  is  well  to  keep  slightly 
within  the  actual  quantities  on  account  of  the  greater  cost  and  labor 
required  to  finish  the  work,  which  would  make  the  latter  part 
appear  so  much  less  profitable  to  the  contractor  as  sometimes  to 
induce  a  disposition  to  abandon  the  work  before  completion. 

240.  The  Classification  of  Earthwork.— It  is  customary  to 
group  earthwork  in  excavation,  according  to  the  difficulty  of 
removal,  into  three  classes— earth-excavation,  loose  rock,  and 
solid  rock — though  other  cla.ssifications  are  frequently  made. 

Earth-excavation  includes  all  earth,  sand,  loam,  and  loose 
stones  that  can  be  moved  with  the  plow  and  scraper. 

Loose  rock  includes  all  stones  and  detached  boulders  less  than 
from  1  to  3  cubic  yards  in  size,  and  all  slate,  shale,  or  cemented 
gravel  requiring  the  use  of  the  pick  and  bar,  but  which  may  be 
removed  without  blasting. 

Solid  rock  includes  all  boulders  above  a  certain  size  (usually 
from  1  to  3  cubic  yards,  as  specified)  and  all  rock  masses  that 
cannot  be  removed  without  blasting. 

The  relative  prices  vary,  but  a  ratio  of  1:3:7  will  not  be  far 
from  an  average  for  the  more  common  conditions  arising  in  rail- 
road work. 

The  neces.sity  for  a  correct  classification  is  evident,  and  the 
engineer  should  keep  full  notes,  and  make  careful  measurements 
whenever  a  given  volume  involves  more  than  one  class  of  earth- 
work. It  is  customary  to  specify  that  his  decision  is  final,  and 
therefore  his  measurements  should  be  cjxrefully  made  during  the 
progress  of  the  work,  and  notes  on  the  nature  of  material  taken 
at  the  same  time. 

A  notebook  should  be  kept  showing,  the  measurements  and 
amounts  of  each  class  of  material  for  each  station,  together  with 
the  date  of  completion  and  acceptance. 


CONSTRUCTION".  207 

241.  A  Progress  Profile  shoultl  accompauy  the  monthly  esti- 
mate to  exhibit  graphically  the  amount  of  work  done  during  the 
month,  different  colors  being  used  for  the  different  mouths.  The 
final  profile  should  show  approximately  the  progress  of  the  work. 
The  colors  may  be  laid  on  witii  a  brush,  or  hatchings  made  with  a 
pen;  in  neither  case  should  the  color  obscure  the  lines  of  the  pro- 
file-paper. A  duplicate  progress  profile  shoultl  be  retained  in  the 
division  engineer's  office;  if  transparent  profile-paper  is  employed, 
one  may  be  simply  traced  through  from  the  other.  A  further 
advantage  of  the  transparent  paper  is  that  blue-prints  of  any  por- 
tion of  the  profile  may  be  readily  made  wneu  duplicates  are 
desired,  provided  the  drawings  are  in  black  or  any  color  admit- 
ting blue  printing. 

242.  Masonry  is  to  be  measured  in  cubic  yards,  and  any 
material  on  hand,  but  not  in  place,  is  to  be  measured  and  esti- 
mated. The  classification  of  masonry  must  be  according  to 
specifications.  Foundation-pits  for  piers  or  culverts  must  be 
measured  as  soon  as  completed,  and  before  the  masonry  has  been 
put  in  place. 

243.  Bridges  must  be  estimated  by  measurement,  or  by 
checking  up  material  in  place  and  that  on  hand  but  not  in  place. 

For  trestle  bridges,  or  foundations  requiring  piling,  the  actual 
number  of  linear  feet  below  cap  must  be  measured;  this  neces- 
sitates the  constant  supervision  of  the  engineer  or  an  assistant, 
sometimes  known  as  a  "pile-recorder,"  whose  duty  it  is  to  see 
that  all  piles  come  up  to  specifications  and  are  driven  in  accord- 
ance therewith. 

All  framing-timber  in  place,  or  delivered  but  not  in  place,  is 
to  be  included  in  the  estimate,  the  amount  being  obtained  by 
measurement. 

Steel  spans  or  trestles  are  to  be  estimated,  in  the  same  manner 
as  wooden  trestles,  by  checking  up  or  measuring  the  material  on 
hand  and  in  place. 

244.  Track  Material  must  be  checked  up  either  by  the"  mate- 
rial clerk  "  or  the  engineer  in  charge  of  track.  Ballasting  prop- 
erly belongs  with  the  graduation,  but  may  be  put  in  place  after 
the  rails  have  been  laid;  in  either  case  it  is  estimated  in  accord- 
ance with  the  specifications. 

For  preliminary  and  monthly  estimates  it  will  be  suflficient  to 


208      A    FIELD-MAXUAL    FOR    RAILROAD    EXGIXEERS. 

estimate  track  material  by  means  of  tables  showing  the  numl)er 
of  cross  ties  for  a  given  spacing  and  the  weight  of  steel  fur  a 
given  rail  section,  but  before  the  final  estimate  is  made  all  mate- 
rial must  be  measured  or  counted. 

245.  Blank  Estimate-sheets  are  sent  out  from  the  chief  euiri- 
neer's  office  to  be  filled  out  by  the  engineers  makiug  estimate, 
who  should  retain  a  copy  of  each  estimate  rendered.  On  these 
sheets  should  appear  the  total  quantit}'  estimated,  the  amount  of 
the  last  preceding  estimate,  and  the  estimate  for  the  month,  which 
will  be  the  difference  of  the  other  two. 

The  division  engineer's  estimate  must  show  not  onl}-  the  cj[uan- 
tity  of  material,  but  its  value  in  dollars  and  cents  computed  from 
the  contract  price.  The  footings  of  the  several  columns  then 
serve  as  a  check  upon  each  other. 

246.  The  Monthly  Payments  are  not  made  for  the  full 
amount  estimated,  but  about  15  or  20  per  cent  is  retained  until 
after  the  final  estimate  has  been  made,  in  order  to  insure  the 
completion  of  the  work  by  the  contractor,  and  to  be  used  as  a 
fund  from  which  to  withhold  the  amount  of  damages  provided  in 
the  contract  for  failure  to  comply  with  all  its  provisions. 

247.  Extras  incident  to  minor  changes,  or  to  the  protection  or 
drainage  of  the  work,  are  usually  shown  on  the  final  estimate,  but 
a  better  way  would  be  to  require  the  contractor  to  present  his  bill 
for  extras  at  the  end  of  each  mouth,  and  to  incorporate  them  in 
the  monthly  estimate  when  they  are  just.  The  engineer  should 
take  measurements  upon  any  extra  work  at  the  time  of  its  com- 
pletion, and  should  keep  a  record  thereof.  If  the  extras  are  of  a 
nature  not  admitting  of  measurement,  he  should  note  the  com- 
pensation to  be  allowed  at  the  time  tbe  extra  work  is  done. 

248.  The  Final  Estimate  must  include  all  earthwork  moved, 
all  material  in  bridges,  all  masonry  in  foundations,  culverts,  piers, 
and  tunnels,  and  all  other  material  supplied  or  work  done  in 
compliance  with  the  contract.  The  engineer  should  keep  his 
notes  full  and  complete  during  the  construction  of  the  work,  in 
order  to  be  able  to  meet  the  contractor's  claims  for  extras  or  com- 
plaints as  to  classification.  Any  items  that  may  have  been  over- 
looked in  making  up  the  monthly  estimates  must  be  includec 
here. 


i 


CONSTRUCTION.  209 

249.  Acceptance.  —  Until  the  en,i;inccr  lias  ijioiiounccd  tlic 
work  satisfactory  and  formally  accepted  it  the  contractor  is 
liable  for  its  couditiou,  aud  must  make  good  all  damage  caused 
by  accident  or  storm.  The  road-bed  and  track  may  be  accepted 
without  special  test;  but  all  spans  should  be  subjected  to  a  speci- 
fied test -load,  under  which  they  must  show  not  more  thari  a  cer- 
tain maximum  deflection,  so  their  acceptance  will  come  last. 

Sometimes  the  contract  requires  a  particular  structure  or  class 
of  structures  to  be  maintained  in  good  order  for  a  certain  length 
of  time  after  completion,  aud  a  percentage  is  retained  to  cover 
the  case. 

After  tinal  acceptance  the  work  is  paid  for  in  accordance  with 
the  final  estimate. 


TABLES. 


ii 


010 

TABLE  I. 

RADII. 

Beg. 

Radius. 

Deg. 

Radius. 

Deg. 

1 
Radius.  \ 

Deg. 

Radius. 

Deg. 

Radius. 

0°    0' 

j 
Infinite 

1°    0' 

5729.65 

2°    0' 

2864.93 

3°    0' 

1910.08 

4°    0' 

14.32.69 

1 

343775. 

1 

56:35.72 

1 

2841.26 

1 

1899.53 

1 

1426.74 

2 

171887. 

2 

5544.83 

2 

2817.97, 

2 

1889.09 

o 

1420.85 

3 

114592.1 

S 

5456.82 

3 

2795.06 

3 

1878.77 

3 

1415.01 

4 

85943.7! 

4 

5371.56 

4 

2772.53 

4 

1868.56 

4 

1409.21 

5 

68754.9 

5 

5288.92 

5 

2750.35 

5 

1858.47 

5 

1403.46 

6 

57295.8 

6 

5208.79 

6 

2728.52 

6 

1848.48 

6 

1397.76 

t 

49110.71 

4 

5131.05 

7 

2707.04 

7 

18:38.59 

1:392.10 

8 

42971.8! 

8 

5055.59 

8 

2685.89 

8 

1828.82 

8 

1:386.49 

9 

33197. 2i 

9 

4982.33 

9 

2665.08 

9 

1819.14 

9 

1380.92 

10 

34377.5 

10 

4911.15 

10 

2644.58 

10 

1809.57 

10 

1375.40 

11 

31252.3 

11 

4841.98 

11 

2624.39 

11 

1800.10 

11 

1:369.92 

12 

28647.8 

12 

4774.74 

12 

2604.51 

12 

1790.73 

12 

1364.49 

13 

26444.2 

13 

4709.  as 

13 

2584.93 

13 

1781.45 

13 

1359.10 

14 

24555.4 

14 

4645.69 

14 

2565 . 65 

14 

1772.27 

14 

1:353.75 

15 

22918.3 

15 

4583.75 

15 

2546.64 

15 

1763.18 

15 

1348.45 

16 

21485.9 

16 

4523  44 

16 

2.527.92 

16 

1754.19 

16 

1:34:3.15 

17 

20222.1 

17 

4464.70 

17 

2509.47, 

17 

1745.26 

17 

1.337.65 

18 

19098  6 

18 

4407.46; 

18 

2491.29 

18 

1736.48 

18 

1.3.32.77 

19 

18093.4 

19 

4351.67] 

19 

2473.  :37 

19 

1727.75 

19 

1:327.63 

20 

17188.8 

20 

4297.28 

20 

2455.70 

20 

1719.12 

20 

1322.53 

21 

16370.2 

21 

4244.23 

21 

24.38.29 

21 

1710  56 

2! 

1317.46 

22 

15626.1 

22 

419-^.47 

22 

2421.12 

22 

1702.10 

22 

1312.43 

23 

14946.7 

23 

4141.96! 

23 

2404.19 

1       23 

1693.72 

23 

1307.45 

24 

14323.6; 

24 

4092.66 

24 

2:387.. 50 

24 

16,S5.42 

24 

1:302. 50 

25 

13751  0 

25 

4044.51 

25 

2:371.04 

25 

1677.20 

25 

1297.58 

26 

13222. ll 

26 

3997.49 

26 

2354.80 

26 

1669.06 

26 

1292.71 

27 

127.32.4 

27 

3951.54 

27 

2:3:38.78 

27 

1661.00 

27 

1287.87 

28 

12277.7 

28 

3906. 54( 

28 

2:322.98 

28 

1653.01 

28 

1283.07 

29 

11854.31 

29 

3862.74 

29 

2307.  :39 

29 

1645.11 

29 

1278.30 

30 

11459.2 

30 

3819.83 

30 

2292.01 

1 

30 

1637.28 

30 

1273.57 

31 

11089  6, 

31 

3777 . 85 

31 

2276.841 

31 

1629.52 

31 

1268.87 

32 

1074:i  0 

32 

3736.79 

32 

2261.86 

32 

1621.84 

32 

1264.21 

33 

10417.5; 

33 

3696  61 

33 

2247.08 

33 

1614.22 

33 

1259.58 

34 

10111.1 

34 

3657.29 

ai 

22:32. 49  i 

34 

1606.68 

34 

1254.98 

35 

9822.18; 

3o 

3618.80, 

a5     2218.09 

1       35 

1599.21 

35 

1250.42 

36 

9549. 34 1 

36 

.3581.10 

36 

2203.87 

!       36 

1591.81 

36 

1245.89 

37 

9291.29' 

'       37 

3544.19! 

37 

2189.84, 

1       37 

1584.48 

37 

1241.40 

38 

9046.75 

38 

:3508.02 

38 

2175.98' 

■       38 

1.577.21 

38 

1236.94 

39 

&S14.78: 

39 

3472.. 59 

39 

2162.30 

39 

1.570.01 

39 

12.92.51 

40 

8594.42 

40 

3437.87 

40 

2148.79, 

1 

40 

1562.88' 

40 

1228.11 

41 

a384.80 

41 

3403.83 

41 

2135.44 

41 

1.555.81 

41 

1223.74 

42 

8185.16, 

42 

3370.46 

42 

2122.26 

1       42 

1.548.80 

42 

1219.40  1 

43 

7994.81! 

43 

33.37.74 

43 

2109.24 

43 

1541.86 

43 

1215.30 

44 

7813. ii; 

44 

:3305.65 

44 

2096.  :39 

!       44 

1.5:M.98 

44 

1210.82 

45 

7639.49! 

45 

3274.17 

45 

2083.68 

45 

1528.16 

45 

1206.. 57 

46 

7473. 42 1 

46 

3243.29 

46 

2071.13 

46 

1.521.40 

46 

1202.36 

47 

7314.41! 

47 

3212.98 

47 

2058.73 

47 

1.514.70 

47 

1198.17 

48 

7162.03 

48 

3183.23 

48 

2046.48 

48 

1508.06 

48 

1194.01 

49 

7015.87 

49 

3154.03 

49 

20:34.  :37 

1       "^9 

1.501.48 

49 

1189.88 

50 

6875 . 55 

50 

3125.36 

i 

50 

2022.41 

50 

1494.95 

50 

1185.78 

51 

6740.74 

51 

3097.20! 

51 

2010.59 

51 

1488.48 

51 

1181.71 

52 

6611.121 

52 

3069.55 

52 

'  199S.90 

52 

1482.07 

52 

1177.66 

53 

6486.38 

':        53 

3042.39 

53 

1987.35 

53 

1475.71 

53 

1173.65 

54 

6366.26, 

54 

,  3015.71 

54 

1  1975.93 

54 

1469.41 

.54 

1169.66 

55 

6250.51' 

55 

;  2989.48 

55 

1964.64 

55 

1463.16 

55 

1165.70 

56 

6138.90, 

56 

2963.71 

56 

1953.48 

56 

1456.96 

56 

1161.76 

57 

6031.20 

57 

29:38.39 

57     1942.44 

57 

1450.81 

57 

11.57. a5 

58 

5927.22 

58 

2913.49 

58     19:31.. 53 

;     58 

1444.72 

58 

11.53.97 

59 

.5826.76 

59 

2889.01 

59     1920.75 

59 

1438.68 

59 

1150.11 

60 

1 

5729.65 

60 

,  2864-93 

60     1910.08 

!       60 

1432.69 

60 

1146.28 

TABLES. 


213 


TABLE 

I.  RADII. 

Deg. 

6°  0' 

Radius. 
1146.28 

[Deg. 

Radius. 

Deg. 

Radius. 

818.64 

Deg. 

Radius. 

Deg. 

Radius. 

6°0' 

955.37 

70  0' 

8°0' 

716.34 

9°0' 

636.78 

I 

1142.47 

1 

952.72 

1 

816.70 

1 

714.85 

1 

635.61 

2 

1138.69' 

2 

950.09 

2 

814.76 

2 

713.37 

2 

634.44 

3 

1134.94 

3 

947.48 

3 

812.83 

3 

711.90 

3 

633.27 

4 

1131.21 

4 

944.83 

4 

810.92 

4 

710.43 

4 

632.10 

5 

1127.50 

5 

942.29 

5 

809.01 

5 

708.96 

5 

630.94 

6 

1123.82 

6 

939.72 

6 

807.11 

6 

707.51 

6 

629.79 

7 

1120.16 

1 

937.16 

7 

805.22 

7 

706.05 

7 

628.04 

8 

1116.52 

8 

934.62 

8 

803.34 

8 

704.60 

8 

627.49 

9 

1112.91 

9 

932.09 

9 

801.47 

9 

703.16 

9 

626.35 

10 

1109.33, 

10 

929.57 

10 

799.61 

10 

701.73 

10 

625.21 

11 

1105.76 

11 

927.07 

11 

797.75 

11 

700.30 

11 

624.08 

12 

1102.22 

12 

924.58 

12 

795.91 

12 

698.88 

12 

622.95 

13 

1098.70 

13 

922.10 

13 

794.07 

13 

697.46 

13 

621.82 

14 

1095.20: 

14 

919.64 

14 

792.24 

14 

696.05 

14 

620.70 

15 

1091.73 

15 

917.19 

15 

790.42 

15 

694.65 

15 

619.58 

16 

1088.28 

16 

914.75 

16 

788.61 

16 

693.24 

16 

618.47 

17 

1084.85 

17 

912.33 

17 

786.80 

17 

691.85 

17 

617.36 

18 

1081.44 

18 

909.92 

18 

785.01 

18 

690.46 

18 

616.25 

19 

1078.05 

19 

907.52 

19 

783.22 

19 

689.08 

19 

615.15 

20 

1074.68 

20 

905.13 

20 

781.44 

20 

687.70 

20 

614.05 

21 

1071.34 

21 

902.76 

21 

779.67 

21 

686.33 

21 

612.96 

22 

1068.01 

22 

900  40 

22 

777.91 

22 

684,96 

22 

611.87 

23 

1064.71 

23 

898.05 

23 

776.15 

23 

683.60  • 

23 

610.78 

24 

1061.43 

24 

895.71 

24 

774.40 

24 

682.25 

24 

609.70 

25 

1058.16 

25 

893.39 

25 

772.66 

25 

680.89 

25 

608.62 

26 

1054.92 

26 

891.08 

26 

770.93 

26 

679.55 

26 

607.55 

27 

1051.70 

27 

888.78 

27 

769.21 

27 

678.21 

27 

606.48 

28 

1048. 48i 

28 

886.49 

28 

767.49 

2& 

676.88 

28 

605.41 

29 

1045.31 

29 

884.21 

29 

765.78 

29 

675.54 

29 

604.35 

30 

1042.14 

30 

881.95 

30 

764.08 

30 

674.22 

30 

603.29 

31 

1039.00' 

31 

879.69 

31 

762.39 

31 

672.90 

31 

602.23 

32 

1035.87 

32 

877.45 

32 

760.70 

32 

671.59 

32 

601.18 

33 

1032.76 

33 

875.22 

33 

759  02 

33 

670.28 

33 

600.13 

34 

1029.671 

34 

873.00 

34 

757.35 

34 

668.98 

34 

599.09 

35 

1026.60 

35 

870.80 

35 

755.69 

35 

667.68 

35 

598.04 

36 

1023.55; 

36 

868.60 

36 

754.03 

36 

666.39 

30 

597.01 

37 

1020.51 

37 

866.41 

37 

752.38 

37 

665.10 

37 

595.97 

38 

1017.49 

38 

864.24 

38 

750.74 

38 

663.82 

38 

594.94 

39 

1014.50 

39 

862.08 

39 

749.10 

39 

662.54 

39 

593  91 

40 

1011.51 

40 

859.92 

40 

747.48 

40 

661.26 

40 

592.89 

41 

1008.55 

41 

857.78 

41 

745.86 

41 

659.99 

41 

591.87 

42 

1005.60 

42 

855.65 

42 

744.24 

42 

658.73 

42 

590.85 

43 

1002.67 

43 

853.53 

43 

742.63 

43 

657.47 

43 

.589.84 

44 

999.76 

44 

851.42 

44 

741.03 

44 

656.22 

44 

588.83 

45 

996.87 

45 

849.32 

45 

739.44 

45 

654.97 

45 

587.83 

46 

993.99 

46 

847.23 

46 

737.86 

46 

653.72 

46 

586.82 

47 

991.13 

47 

845.15 

47 

736.28 

47 

652.48 

47 

585.83 

48 

988.28 

48 

843.08 

48 

734.70 

48 

651.25 

48 

584.83 

49 

985.45 

49 

841.02 

49 

733.14 

49 

650.02 

49 

583.84 

50 

982.64 

50 

838.97 

50 

731.58 

50 

648.79 

50 

582.85 

51 

979.84 

51 

836.93 

51 

730.03 

51 

647.57 

51 

581.86 

52 

977.06 

52 

834.90 

52 

728.48 

52 

646.35 

52 

580.88 

53 

974.29 

53 

832.89 

53 

726.94 

53 

645.14 

53 

579.90 

54 

971.54 

54 

830.88 

54 

725.41 

54 

643.94 

54 

578.92 

55 

968.81 

55 

828.88 

55 

723.88 

55 

642.73 

55 

577.95 

56   966.09 

56 

826.89 

56 

722.36 

56 

641.53 

56 

576.98 

57   963.39 

57 

824.91 

57 

720.85 

57 

640.34 

57 

576.02 

58 

9G0.70 

58 

822.93 

58 

719.34 

58 

639.15 

58 

575.06 

59 

958.03 

59 

820.97 

59 

717.84 

59 

637.06 

59 

574.10 

60   955.37 

60 

819.02 

GO 

716.34 

60 

636.78 

60 

573.14 

214     A   FIELD-MANUAL    FOR   RAILROAD    ENGINEERS. 


TABLE   I.— RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

10°  0 

'  573.14 

12°  0 

477.68 

14°  0' 

409.32 

16°  0' 

358.17 

18°  0' 

318.39 

2 

571.24 

2 

476.36 

o 

'Si' 

408.35 

2 

357.43 

2 

317.80 

4 

569.35 

4 

475.05 

4 

407.38 

4 

356.69 

4 

317.22 

6 

567.47 

6 

473.74 

6 

406.42 

6 

355.95 

6 

316.63 

8 

565.60 

8 

472.44 

8 

405.46 

8 

355.21 

8 

316.05 

10 

563.75 

10 

471.15 

10 

404.51 

10 

354.48 

10 

315.47 

12 

561.91 

12 

469.86 

12 

403.56 

12 

353.75 

12 

314.89 

14 

560.08 

14 

468.58 

14 

402.61 

14 

353.03 

14 

314.32 

16 

558.26 

16 

467.31 

16 

401.67 

16 

352.. 30 

16 

313.75 

18 

556.45 

18 

466.04 

18 

400.74 

18 

351.58 

18 

313.18 

20 

554.66 

20 

464.78 

20 

399.80 

20 

350.86 

20 

312.61 

22 

552.88 

22 

463.53 

22 

398.88 

22 

350.15 

22 

312.04 

24 

551.11 

24 

462.29 

24 

397.95 

24 

349.44 

24 

311.47 

26 

549.35 

26 

461.05 

26 

397.03 

26 

348.72 

26 

310.91 

28 

547.60 

28 

459.82 

28 

396.13 

28 

348.02 

28 

310.35 

30 

545.87 

30 

458.59 

30 

395.21 

30 

347.. 32 

30 

309.79 

32 

544.14 

32 

457.38 

32 

394.30 

32 

346.62 

32 

309.23 

34 

542.42 

34 

456.16 

34 

393.40 

34 

345.93 

34 

308.68 

36 

540.72 

36 

454.96 

36 

392.50 

36 

345.23 

36 

308.13 

38 

539.03 

38 

453.76 

38 

391.61 

38 

344.54 

38 

307.58 

40 

537.34 

40 

452.57 

40 

390.72 

40 

343.85 

40 

307.03 

42 

5S5 . 67 

42 

451.38 

42 

389.83 

42 

343.16 

42 

306.48 

44 

534.01 

44 

450.20 

44 

388.95 

44 

342.48 

44 

305.93 

46 

532.36 

46 

449.02 

46 

388.07 

46 

341.80 

46 

305.39 

48 

.•530.71 

48 

447.86 

48 

387.20 

48 

341.12 

48 

304.85 

50 

529.08 

50 

446,69 

50 

.386.33 

50 

340.45 

50 

304.31 

52 

527.46 

52 

445.54 

52 

385.47 

52 

339.78 

52 

303.77 

54 

525.85 

54 

444.39 

54 

384.60 

54 

339.11 

54 

303.24 

56 

.524.25 

56 

443.24 

56 

383.75 

56 

338.44 

56 

302.70 

58 

522.65 

58 

442.11 

58 

382.89 

58 

337.77 

58 

302.17 

11°  0' 

521.07 

13°  0' 

440.97 

16°  0' 

382.04 

17°  0' 

337.11 

19°  0' 

301.64 

2 

519.50 

2 

439.85 

2 

381.19 

2 

336.45 

2 

301.12 

4 

517.93 

4 

438.73 

4 

380.35 

4 

335.80 

4 

300.59 

6 

516.38 

6 

■  437.61 

6 

379.51 

6 

335.14 

6 

300.07 

8 

514.84 

8 

436.50 

8 

378.68 

8 

3.34.49 

8 

299.54 

10 

513.30 

10 

4.35.40 

10 

377.84 

10 

333.84 

10 

299.02 

12 

511.77 

12 

434.30 

12 

377.02 

12 

333.19 

12 

298.50 

14 

510.26 

14 

433.21 

14 

376.19 

14 

332.55 

14 

297.99 

16 

.508.75 

16 

432.12 

16 

375.37 

16 

331.91 

16 

297.47 

18 

507.25 

18 

431.04 

18 

374.55 

18 

331.27 

18 

296.96 

20 

505.76 

20 

429.96 

20 

373.74 

20 

330.63 

20 

296.45 

22 

504.28 

22 

428.98 

22 

372.93 

22 

330.00 

22 

295.94 

24 

.502.80 

24 

427.82 

24 

372.12 

24 

329.37 

24 

295.43 

26 

501.34 

26 

426.76 

26 

371.32 

26 

328.74 

26 

294.92 

28 

499.88 

28 

425.71 

28 

370.. 52 

28 

328.11 

28 

294.42 

30 

498.43 

30 

424.66 

30 

369.72 

30 

327.48 

30 

293.91 

32 

496.99 

32 

423.61 

32 

368.93 

32 

326.86 

32 

293.41 

34 

495.56 

34 

422.57 

34 

368.14 

.34 
V36 

326.24 

34 

292.91 

36 

494.14 

36 

421.54 

36 

367.35 

325.62 

36 

292.41 

38 

492.73 

38 

420.51 

38 

366.57 

38 

325.01 

38 

291.92 

40 

491.32 

40 

419.49 

40 

365.79 

40 

324.40 

40 

291.42 

42 

489.92 

42 

418.47 

42 

365.01 

42 

323.79 

42 

290.93 

44 

488.53 

44 

417.45 

44 

364.24 

44 

323.18 

44 

290.44 

46 

487.15 

46 

416.44 

46 

363.47 

46 

322.57 

46 

289.95 

48 

485.77 

48 

415.44 

48 

362.70 

48 

321.97 

48 

289.46 

50 

484.40 

50 

414.44 

50 

361.94 

50 

321.37 

50 

288.98 

52 

483.05 

52 

413.44 

52 

361.18 

52 

320.77 

52 

288.49 

54 

481.69 

54 

412.45 

54 

360.42 

54 

320.17 

54 

288.01 

56 

480.35 

56 

411.47 

56 

.3.59.67 

56 

319.57 

56 

287.53 

58 

479.01 

58 

410.49 

58 

.3.58.92 

58 

318.98 

58 

287.05 

60 

477.68 

60 

409.51 

60 

358.17 

60 

318.39 

60 

286.57 

TABLE  II.— MINUTES  IN  DECIMALS  OF  A  DEGREE.   215 


# 

0* 

10" 

15' 

20" 

SO" 

40" 

45' 

50" 

/ 

0 

.00000 

00278 

.00417 

,005.56 

.00833 

.01111 

.01250 

.01389 

0 

1 

.01667 

.01944 

.020S3 

.02222 

.02.500 

.02778 

.02917 

.03055 

1 

2 

.0333;3 

.03611 

.03750 

.0:3889 

.04167 

.04444 

.0458:3 

.04722 

2 

3 

.05000 

.05278 

.05417 

05556 

.0583:3 

.00111 

.06250 

.06:389 

3 

4 

.06667 

.06944 

.07083 

.07222 

.07500 

.07778 

.07917 

.08056 

4 

5 

.08333 

.08611 

.08750 

.08889 

.09167 

.09444 

.09.583 

.09722 

5 

6 

.10000 

.10278 

.10417 

.10556 

.10833 

.11111 

.11250 

.11:389 

6 

7 

.11667 

.119-W 

.  1208:3 

■  i./V'^'W 

.12500 

.1277'8 

.12917 

.13056 

7 

8 

.13:333 

.13611 

.13750 

.13889 

.14167 

.14444 

. 14583 

.14722 

8 

9 

.15000 

. 15278 

.1.5417 

.  15556 

.158:33 

.16111 

.16250 

.16:389 

9 

10 

.16667 

.16944 

.17083 

.17222 

.17500 

.17778 

.17917 

.18056 

10 

11 

.18333 

.18611 

.  187.50 

.18889 

.19167 

19444 

.19.583 

.19722 

11 

12 

.20000 

.20278 

.20417 

.20556 

.20833 

.21111 

.21250 

.21.389 

12 

13 

.21607 

.21944 

.22083 

.22222 

.22500 

.22778 

.22917 

.23056 

13 

14 

.23333 

.23611 

.2:3750 

.2:3889 

.24167 

.24444 

.24583 

.24722 

14 

15 

.25000 

.252;8 

.25417 

.25556 

.25833 

.26111 

.26250 

.26389 

15 

16 

.26007 

.26944 

.27083 

.27222 

.27500 

.2777'8 

.27917 

.28056 

16 

17 

.28333 

.28011 

.28750 

.28889 

.29167 

.29444 

.29583 

.29722 

17 

18 

.30000 

.30278 

.30417 

.30556 

.:30833 

.31111 

.312.50 

.31:389 

18 

19 

.31667 

.31944 

.:32083 

..32500 

.32778 

.32917 

.33056 

19 

20 

.;33333 

.3:3611 

.33750 

.33889 

.34167 

.34444 

.34583 

.34722 

20 

21 

.35000 

.3,5278 

.35417 

.355,56 

..35833 

.36111 

.36250 

.36389 

21 

.36067 

.36944 

.37083 

.37222 

.37500 

.3777'8 

.37917 

.38056 

22 

23 

.383:3:3 

.38611 

.:38r.50 

.38889 

.39167 

.39444 

.39583 

.39722 

23 

24 

.40000 

.40278 

.40417 

.40556 

.408:33 

.41111 

.41250 

.41389 

24 

25 

.41667 

.41944 

.42083 

.42222 

.42500 

.4277'8 

.42917 

.4:3056 

25 

26 

.43:333 

.4:3611 

.43750 

.43889 

.44107 

.44444 

.44583 

.44722 

26 

27 

.4.5000 

.45278 

.45417 

.45556 

.45833 

.46111 

.46250 

.46389 

27 

28 

.46667 

.46944 

.47083 

.47222 

.47500 

.  47'77'8 

.47917 

.48056 

28 

29 

.48333 

.48611 

.48750 

.48889 

.49167 

.49444 

.49583 

.49722 

29 

30 

..50000 

.50278 

.50417 

.50556 

.50833 

.51111 

.51250 

.51389 

30 

31 

..5166? 

.5i941 

.,52083 

.52222 

.52500 

.52778 

.52917 

.53056 

31 

32 

..5:3333 

.5:3011 

..537,50 

..53889 

.54167 

.54444 

.54583 

.54722 

32 

33 

..5.5000 

.5.5278 

.5.5417 

.55556 

.55833 

.56111 

.56250 

.56389 

33 

34 

..56667 

.56944 

.57083 

.57222 

.57500 

.57778 

.57917 

.58056 

;34 

35 

..58:3:33 

..58011 

..5S7.50 

..58889 

.59167 

.59444 

.59583 

.59722 

35 

36 

.600eX) 

.00278 

.60417 

.605.56 

.fi0833 

.61111 

.61250 

.61389 

36 

37 

.01667 

.()1944 

.62083 

.62222 

.02500 

.62778 

.62917 

.6:3056 

37 

38 

.0:3:3:3:3 

.0:3011 

.6:37.50 

.6:3889 

.64167 

.64444 

.64583 

.64722 

.38 

39 

.6.5000 

.6.5278 

.6.5417 

.65556 

.65833 

.66111 

.662.50 

.66:389 

39 

40 

.06667 

.66944 

.67083 

.67222 

.67500 

.67778 

.67917 

.68056 

40 

41 

.68.3:33 

.68611 

.68750 

.68889 

.69167 

.69444 

.69583 

.69722 

41 

42 

.70000 

.70278 

.70417 

.70556 

.70833 

.71111 

.71250 

.71.389 

42 

4:3 

.71607 

.71944 

.7208:3 

.72222 

.72500 

.72778 

.72917 

.73056 

43 

44 

.73:3:33 

.7:3611 

.7.3750 

.73889 

.74167 

.74444 

.74583 

.74722 

44 

45 

.7.5000 

.7.5278 

.75417 

.75556 

.758.33 

.76111 

.76250 

.76389 

45 

46 

.76607 

.76944  1 

.77083 

.77222 

.77500 

.7777'8 

.77917 

.780.56 

46 

47 

.78:3:33 

.78611 

.78750 

.78889 

.79167 

.79444 

.79583 

.79722 

47 

4S 

.80000 

.80278  ; 

.804K»< 

.80556 

.808:33 

.81111 

.81250 

.81.389 

48 

49 

.81667 

.81944 

.82083 

.82222 

.82.500 

.82778 

.82917 

.83056 

49 

50 

.  8:3:3:33 

.83611 

.8:3750 

.83889 

.84167 

.84444 

.84583 

.84722 

50 

51 

.8.5000 

.85278 

.8,5417 

.a5556 

.85833 

86111 

.86250 

.86389 

51 

52 

.80007 

.86944 

.87083 

.87222 

.87500 

.87  < 78 

.87917 

.880.56 

.52 

53 

.88:3:33  1 

.88611 

.88750 

.88889 

.89167 

.89444 

.89583 

.89722 

53 

54 

.90000  i 

.90278 

.90417 

.90.556 

.90ft:33 

.91111 

.91250 

.91:389 

54 

55 

.91667 

.91944 

.92083 

92222 

.92.500 

.92778 

.92917 

.93056 

55 

56 

.9.3.3.3:3 

.9:3611 

.9.37.50 

! 93889 

.94167 

.94444 

.94583 

.94722 

56 

57 

.9.5000 

.9.5278 

.9.5417 

.9.5.5.56 

.95^33 

.96111 

.962.50 

.96:389 

57 

58 

.96007 

.96944 

.970a3 

.97222 

.97.500 

.97778 

.97917 

.98056 

58 

59 

/ 

.98333 

.98611 

.96750 

.98889 

.99167 

.99444 

.99583 

.99722 

59 

0" 

10"   1 

15-   1 

SO- 

30" 

40° 

45" 

50- 

/ 

21G      A    FIELD-MANUAL    FOR    RAILROAD    EKGIXEERS. 


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TABLES. 


17 


TABLE   IV.- 

■LONG 

CHORDS. 

T-\    .  ___. 

^  _     _  .e 

Actual  Arc. 

Long  Chord 

s. 

Depfree  oi 

Curve. 

One  Station. 

o 

3 

4 

5 

6 

stations 

Stations 

.  Stations. 

Stations. 

Stations. 

0° 

10' 

100.000 

200.00 

300.00 

400.00 

500.00 

599.99 

20 

.000 

200.00 

300.00 

399.99 

499.98 

.599.97 

30 

.000 

200.00 

299.99 

399.98 

499.96 

.599.93 

40 

.001 

200.00 

299.99 

399.97 

499.93 

599.88 

50 

.001 

200.00 

299.98 

399.95 

499.89 

.599.82 

1 

100.001 

199.99 

299.97 

399.92 

499.85 

599.73 

10 

.002 

199.99 

299.96 

399.90 

499.79 

.599.64 

20 

002 

199.99 

299.95 

399.87 

499  73 

599.. 53 

30 

.003 

199.98 

299. 93 

399.83 

499.66 

599.40 

40 

.003 

199.98 

299.92 

399.79 

499.. 58 

.599.26 

50 

.004 

199.97 

299.90 

399.74 

499.49 

599.11 

2 

100.005 

199.97 

299.88 

399.70 

499  39 

.598.93 

10 

.006 

199.96 

299.86 

399.64 

499.29 

.598.75 

20 

.007 

199.96 

299.83 

399.. 59 

499.17 

598.. 55 

80 

.003 

199.95 

299.81 

399.52 

499.05 

.598.34 

40 

.009 

199.95 

299.78 

399.46 

498.92 

598.11 

50 

.010 

199.94 

299.76 

399.39 

498.78 

597.86 

3 

100,011 

199.93 

299.73 

399.32 

498.63 

597.60 

10 

.013 

199.92 

299.70 

399.24 

498.47 

597.33 

20 

.014 

199.92 

299.66 

399.15 

498.31 

597.04 

30 

.015 

199.91 

299.63 

399.07 

498.14 

596.74 

40 

.017 

199.90 

299.. 59 

398.98 

497.96 

.596.42 

50 

.019 

199.89 

299.55 

398.88 

497.77 

596.09 

4 

100.020 

199.88 

299.51 

398.78 

497  57 

.595.74 

10 

.022 

199.87 

299.47 

398.68 

497.36 

595.38 

20 

.024 

199.86 

299.43 

398.. 57 

497.15 

595.01 

30 

.0-'6 

] 99 . 85 

299.38 

398.40 

496.92 

594.62 

40 

.028 

199.83 

299.34 

398.34 

496.69 

•594.21 

50 

.030 

199.82 

299.29 

398  22 

496.45 

593.79 

5 

100.032 

199  81 

299.24 

.398.10 

496.20 

593.36 

10 

.034 

199.80 

299.19 

397.97 

495.94 

.592.91 

20 

.036 

199  78 

299.13 

397.84 

495.68 

592.45 

30 

.038 

199.77 

299.08 

397.70 

495.41 

591.97 

40 

.041 

199.76 

299.02 

397.56 

495.12 

591.48 

50 

.043 

199.74 

298.96 

397.41 

494.83 

590.97 

G 

100.046 

199.73 

298.90 

397. 2G 

494.53 

590.45 

10 

.04S 

199.71 

298.84 

397.11 

494.23 

.589.91 

20 

.051 

199.70 

298.78 

396.95 

493.91 

589.36 

•30 

.054 

199.68 

298.71 

396.79 

493.59 

588.80 

40 

.056 

1!)9.G6 

298.65 

396.62 

493.26 

588.22 

50 

.059 

199.64 

298.. 58 

.396.45 

492.92 

587.63 

7 

100.062 

199.63 

298.51 

396.28 

492.57 

587.02 

10 

.016 

199.51 

298.. 30 

395  91 

491.97 

.586.12 

20 

.017 

199.49 

298.21 

395.71 

491.60 

585.46 

30 

.018 

199.47 

298.13 

395.. 52 

491.21 

.584.80 

40 

.018 

199.44 

298.05 

.395.32 

490.82 

.584.13 

50 

.019 

199.42 

297.96 

395.11 

490.42 

583.44 

S 

100.020 

199.39 

297.87 

394.90 

490.01 

582.72 

10 

.021 

199.37 

297.78 

394.69 

489.59 

582.01 

20 

.022 

199.31 

297.69 

394.47 

489.16 

.581.27 

30 

.023 

199.31 

297.60 

.394.25 

488.73 

.580.52 

40 

.024 

199.28 

297.50 

394.02 

488.28 

.579.76 

50 

.025 

199.26 

297.41 

.393.79 

487.83 

578.98 

9 

100.026 

199.23 

297.31 

393.. 55 

487.37 

.578.18 

10 

.027 

199  20 

297  21 

393.31 

486.90 

.577.38 

20 

.028 

199.17 

297.10 

393.07 

486.43 

576.. 56 

•30 

.029 

199.  U 

297.00 

.392.82 

485.95 

.575.73 

40 

.030 

199  11 

296.90 

392.. 57 

485.45 

574.88 

50 

.031 

1 99 . 08 

296.79 

392.31 

484.95 

.574.02 

10 

100.032 

199.05 

296.68 

392.05 

484.44 

573.14 

218     A    FIELD-MANUAL   FOR    RAILROAD    ENGINEERS. 


TABLE  IV.— LONG  CHORDS. 


Degree 

of 
Curve. 


10° 


11 


12 


13 


14 


0' 
10 
■20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 


Actual  Arc, 

One 

Station. 


100.0.32 
.033 
.034 
.035 
.036 
.037 

100.038 
.040 
.041 
.042 
.043 
.044 

100.046 
.047 
.048 
.050 
.051 
.052 

100.0.54 
.055 
.056 
.058 
.059 
.061 

100.062 


Long  Chords. 


12  3  4  5 

Station.     Stations.  Stations.   Stations.  Stations. 


99.91 
99.90 
99.90 
99.90 
99.89 
99.89 
99.88 
99.88 
99.88 
99.87 
99.87 
99.87 

99.86 
99.86 
99.85 
99  85 
99.85 
99.84 
99.84 
99.84 
99.83 
99.83 
99.82 
99.82 
99.81 


199  05 
199.02 
198.98 
198.95 
198.92 
198.88 
198.85 
198.81 
198  78 
198.74 
198.71 
198.67 

198.63 
198.. 59 
198.55 
198.51 
198.48 
198.43 
198.39 
198.35 
198.31 
198.27 
198.23 
198.18 
198.14 


296.68 
296.57 
296.45 
296.33 
296.22 
296.10 
295,98 
295.86 
295.74 
295.61 
295.48 
295.35 

295.22 
295.09 
294.95 
294.82 
294.68 
294.54 
294.40 
294.26 
294.11 
293.96 
293.81 
293.66 
293.51 


392.05 
.391.79 
391.51 
391.24 
390.96 
390.68 
390.39 
390.10 
389.81 
389.. 50 
389.20 
388.89 

388.58 
388.27 
387.95 
387.62 
387.29 
386.95 
386.62 
386.28 
385.93 
385.58 
385.23 
384.87 
384.. 51 


484.44 
483.92 
483.39 

482.86 
482.32 
481.77 
481.21 
480.64 
480.07 
479.48 
478.89 
478.29 

477.68 
477.07 
476.44 
475.81 
475.18 
474.52 
473.87 
473.20 
472.53 
471.86 
471.17 
470.48 
469.77 


TABLE   v.— MID-ORDINATES   TO   LONG   CHORDS. 


Degree  of 

1 

o 

3 

4 

5 

...     ., 
6 

Curve. 

Station. 

Stations. 

Stations. 

Stations. 

Stations. 

Stations. 

0°  10' 

.04 

.15 

.33 

.58 

.91 

1.31 

20 

.07 

.29 

.65 

1.16 

1.82 

2.62 

30 

.11 

.44 

.98 

1.75 

2.73 

8.93 

40 

.15 

.58 

1.31 

2  33 

3.64 

5.24 

50 

.18 

.73 

l.tj4 

2.91 

4.55 

6.54 

1 

.22 

.87 

1.96 

3.49 

5.45 

7.85 

10 

.26 

1.02 

2.29 

4.07 

6.36 

9.16 

20 

.29 

1.16 

2.62 

4.65 

7.27 

10.47 

30 

.33 

1.31 

2.95 

5.24 

8.18 

11.78 

40 

.36 

1.45 

3.27 

5.83 

9.09 

13.08 

50 

.40 

1.60 

3.60 

6.40 

9.99 

14.39 

9 

.44 

1.75 

3.93 

6.98 

10.90 

15  69 

10 

.47 

1.89 

4.25 

7.56 

11.81 

17.00 

20 

.51 

2.04 

4.58 

8.14 

12.72 

18.30 

30 

.55 

2.18 

4.91 

8.72 

13.62 

19.61 

40 

.58 

2.33 

5.23 

9.30 

14.53 

20.91 

50 

.62 

2.47 

5.56 

9.88 

15.44 

22.21 

3 

.65 

2.62 

5.89 

10.46 

16.34 

23.52 

10 

.69 

2.76 

6.22 

11.04 

17.25 

24.82 

20 

.73 

2.91 

6.54 

11.62 

18.15 

26.12 

30 

.76 

3.05 

6.87 

12.20 

19.06 

27.42 

40 

.80 

3.20 

7.20 

12.78 

19.96 

28.71 

50 

.84 

3.35 

7.52 

13.. 36 

20.86 

30.01 

4 

.87 

3.49 

7.85 

13.94 

21.77 

31.31 



TABLES. 


219 


TABLE   v.— MID-ORUINATES   TO   LONG   CHORDS. 


Degree  of 

1 

2 

3 

4 

5 

6 

Curve. 

Station. 

Stations. 

Stations. 

Stations. 

Stations. 

Stations. 

4» 

0' 

.87 

3.49 

7.85 

13.94 

21.77 

31.31 

10 

.91 

3.64 

8.18 

14.52 

22.67 

32  60 

20 

.95 

3.78 

8.50 

15.10 

23.57 

33.90 

30 

.98 

3.93 

8.83 

15.68 

24.47 

35.19 

40 

1.02 

4.07 

9.15 

16.26 

25.37 

36.48 

50 

1.05 

4  22 

9.48 

16.84 

26.27 

37.77 

5 

1.09 

4.36 

9.81 

17.42 

27.17 

39.06 

10 

1.13 

4.51 

10.13 

17.99 

28.07 

40.35 

20 

I.IG 

4.65 

10.46 

18.57 

28.97 

41.63 

30 

1.20 

4.80 

10.79 

19.15 

29.87 

42.92 

40 

1  24 

4.94 

11.11 

19.72 

30.76 

44.20 

50 

1.27 

5.09 

11.44 

20.30 

31.66 

45.48 

6 

1.31 

5  23 

11.76 

20.88 

32.55 

46.76 

10 

1.35 

5.38 

12.09 

21.45 

.33.45 

48.04 

20 

1.38 

5.52 

12.41 

22  03 

34.34 

49.31 

30 

1.42 

5.67 

12.74 

22.60 

35.23 

50.59 

40 

1.16 

5.81 

13.06 

23.18 

36.13 

51.86 

50 

1.49 

5.96 

13.39 

23.75 

37.02 

53.13 

7 

1  53 

6.11 

13.72 

24.33 

37.91 

.54.40 

10 

1.56 

6.25 

14.03 

24.89 

38.79 

55.64 

20 

1.60 

6.39 

14  36 

25.46 

39.66 

56.90 

30 

1  64 

6.54 

14.68 

26.04 

40.55 

58.16 

40 

1.67 

6.68 

15.01 

26.61 

41.43 

59.41 

.50 

1.71 

6.83 

15.33 

27.18 

42.32 

60.68 

8 

1.74 

6.97 

15.65 

27.75 

43.20 

61.93 

10 

1.78 

7.12 

15.98 

28.32 

44.08 

63.18 

20 

1.82 

7.26 

16.30 

28.89 

44.96 

64.43 

30 

1.85 

7.41 

16.62 

29.46 

45.84 

65.68 

40 

1.89 

7.55 

16.95 

30.03 

46  72 

66.92 

50 

1.93 

7.70 

17.27 

30.60 

47.60 

68.17 

9 

1.96 

7.84 

17.59 

31.17 

48.47 

69.40 

10 

2.00 

7.98 

17.92 

31.73 

49.35 

70.64 

20 

2.04 

8.13 

18.24 

32.30 

.50.22 

71.88 

30 

2.07 

8.27 

18.56 

32.87 

51.09 

73.11 

40 

2.11 

8.42 

18  88 

33.43 

51.96 

74.34 

50 

2.14 

8.. 56 

19.21 

34.00 

52.83 

75.56 

10 

2.18 

8.71 

19.53 

.34.56 

53.70 

76.79 

10 

2  22 

8.85 

19.85 

35.13 

.54.57 

20 

2  25 

9.00 

20.17 

35.69 

55.43 

30 

2.29 

9.14 

20.49 

36.26 

56.29 

40 

2.. 33 

9.28 

20.82 

36.82 

57.16 

50 

2.. 36 

9.43 

21.14 

37.38 

58.02 

11 

2.40 

9.57 

21.46 

37.94 

58.88 

10 

2.44 

9.72 

21.78 

38.50 

59.73 

20 

2.47 

9.86 

22.10 

39  00 

60.58 

30 

2.51 

10.01 

22.42 

39.62 

61.44 

40 

2.54 

10.15 

22.74 

40.18 

62.30 

50 

2.58 

10.29 

23.06 

40.74 

63.15 

12 

2  62 

10.44 

23.38 

41.30 

64.00 

10 

2.65 

10.58 

23.70 

41.86 

64.85 

20 

2  69 

10.73 

24.02 

42.41 

65.69 

30 

2.73 

10.87 

24.34 

42.97 

66.54 

40 

2.76 

11.01 

24.66 

43.52 

67.38 

50 

2.80 

11.16 

24.97 

44.08 

68.22 

13 

2.83 

11.30 

25.29 

44.63 

69.06 

« 

10 

2.87 

11.45 

25.61 

45.18 

69.90 

20 

2.91 

11. .59 

25.93 

45.73 

70.73 

30 

2.94 

11.73 

26  25 

46.29 

71.57 

40 

2  98 

11.88 

26.. 57 

46.84 

72.40 

50 

3.02 

12.02 

26.88 

47.39 

73.23 

14 

3  05 

12.16 

27.20 

47  93 

74.06 

1 

N 

O    12   3   4 

5    6   7    8   9 

100 

00000  00043  00087  00130  00173  00217  00260  00303  00346  00389 

1 

0432  0475  0518  0561  0604 

0647  0689  0732  0775  0817 

2 

0860  0903  0945  0988  1030 

1072  1115  1157  1199  1242 

3 

1284  1326  1368  1410  1452 

1494  1536  1578  1620  1662 

4 

1703  1745  1787  1828  1870 

1912  1^53  1995  2036  2078 

5 

2119  2160  2202  2243  2284 

2325  2366  2407  2449  2490 

6 

2531  2572  2612  2653  2694 

2735  2776  2816  2857  2898 

7 

2938  2979  3019  3060  3100 

3141  3181  3222  3262  330^ 

8 

3342  3383  3423  3463  3503 

3543  3583  3623  3663  3703 

9 

3743  3782  3822  3862  3902 

3941  3981  4021  4060  4100 

110 

04139  04179  04218  04258  04297  04336  04376  04415  04454  04493 

1 

4532  4571  4610  4650  4689 

4727  4766  4805  4844  4883 

2 

4922  4961  4999  5038  5077 

5115  5154  5192  5231  5269 

3 

5308  5346  5385  5423  5461 

5500  5538  5576  5614  5652 

4 

5690  5729  5767  5805  5843 

5881  5918  5956  5994  6032 

5 

6070  6108  6145  6183  6221 

6258  6296  6333  6371  6408 

6 

6446  6483  6521  6558  6595 

6633  6670  6707  6744  6781 

7 

6819  6856  6893  6930  6967 

7004  7041  7078  7115  7151 

8 

7188  7225  7262  7298  7335 

7372  7408  7445  7482  7518 

9 

7555  7591  7628  7664  7700 

7737  7773  7809  7846  7882 

120 

07918  07954  07990  08027  08063 

08099  08135  08171  08207  08243 

1 

8279  8314  8350  8386  8422 

8458  8493  8529  8565  8600 

2 

8636  8672  8707  8743  8778 

8814  8849  8884  8920  8955 

3 

8991  9026  9061  9096  9132 

9167  9202  9237  9272  9307 

4 

9342  9377  9412  9447  9482 

9517  9552  9587  9621  9656 

5 

9691  9726  9760  9795  9830 

9864  9899  9934  9968  10003 

6* 

10037  10072  10106  10140  10175  10209  10243  10278  10312  0346  | 

7 

0380  0415  0449  0483  0517 

0551  0585  0619  0653  0687 

8 

0721  0755  0789  0823  0857 

0890  0924  0958  0992  1025 

9 

1059  1093  1126  1160  1193 

1227  1261  1294  1327  1361 

130 

11394  11428  11461  11494  11528  11561  11594  11628  11661  11694 

1 

1727  1760  1793  1826  1860 

1893  1926  1959  1992  2024 

2 

2057  2090  2123  2156  2189 

2222  2254  2287  2320  2352 

3 

2385  2418  2450  2483  2516 

2548  2581  2613  2646  2678 

4 

2710  2743  2775  2808  2840 

2872  2905  2937  2969  3001 

5 

3033  3066  3098  3130  3162 

3194  3226  3258  3290  3322 

6 

3354  3386  3418  3450  3481 

3513  3545  3577  3609  3640 

7 

3672  3704  3735  3767  3799 

3830  3862  3893  3925  3956 

8 

3988  4019  4051  4082  4114 

4145  4176  4208  4239  4270 

9 

4301  4333  4364  4395  4426 

4457  4489  4520  4551  4582 

140 

14613  14644  14675  14706  14737 

14768  14799  14829  14860  14891 

1 

4922  4953  4983  5014  5045 

5076  5106  5137  5168  5198 

2 

5229  5259  5290  5320  5351 

5381  5412  5442  5473  5503 

3 

5534  5564  5594  5625  5655 

5685  5715  5746  5776  5806 

4 

5836  5866  5897  5927  5957 

5987  6017  6047  6077  6107 

5 

6137  6167  6197  6227  6256 

6286  6316  6346  6376  6406 

6 

6435  6465  6495  6524  6554 

6584  6613  6643  6673  6702 

7 

6732  6761  6791  6820  6850 

6879  6909  6938  6967  6997 

8 

7026  7056  7085  7114  7143 

7173  7202  7231  7260  7289 

9 

7319  7348  7377  7406  7435 

7464  7493  7522  7551  7580 

150 

17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 

TABLE  VI.— LOGARITHMS   OF   NUMP.EIJS.  22\ 


N 
150 

0123456789 

17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 

1 

7898  7926  7955  7984  8013  8041  8070  8099  8127  8156 

2 

8184  8213  8241  8270  8298  8327  8355  8384  8412  8441 

3 

8469  8498  8526  8554  8583  8611  8639  8667  8696  8724 

4 

8752  8780  8808  8837  8805  8893  8921  8949  8977  9005 

5 

9033  9061  9089  9117  9145  9173  9201  9229  9257  9285 

6 

9312  9340  9368  9396  9424  9451  9479  9507  9535  9562 

7 

9590  9618  9645  9673  9700  9728  9756  9783  9811  9838 

8 

9866  9893  9921  9948  9976  20003  20030  20058  20085  20112 

9 

20140  20167  20194  20222  20249  0276  0303  0330  0358  0385 

160 

20412  20439  20466  20493  20520  20548  20575  20602  20629  20656 

1 

0683  0710  0737  0763  0790  0817  0844  0871  0898  0925 

2 

0952  0978  1005  1032  1059  1085  1112  1139  1165  1192 

3 

1219  1245  1272  1299  1325  1352  1378  1405  1431  1458 

4 

1484  1511  1537  1564  1590  1617  1643  1669  1696  1722 

5 

1748  1775  1801  1827  1854  1880  1906  1932  1958  1985 

6 

2011  2037  2063  2089  2115  2141  2167  2194  2220  2246 

7 

2272  2298  2324  2350  2376  2401  2427  2453  2479  2505 

8 

2531  2557  2583  2608  2634  2660  2686  2712  2737  2763 

9 

2789  2814  2840  2866  2891  2917  2943  2968  2994  3019 

170 

23045  23070  23096  23121  23147  23172  23198  23223  23249  23274 

1 

3300  3325  3350  3376  3401  3426  3452  3477  3502  3528 

2 

3553  3578  3603  3629  3654  3679  3704  3729  3754  3779 

3 

3805  3830  3855  3880  3905  3930  3955  3980  4005  4030 

4 

4055  4080  4105  4130  4155  4180  4204  4229  4254  4279 

5 

4304  4329  4353  4378  4403  4428  4452  4477  4502  4527 

6 

4551  4576  4601  4625  4650  4674  4699  4724  4748  4773 

7 

4797  4822  4846  4871  4895  4920  4944  4969  4993  5018 

8. 

5042  5066  5091  5115  5139  5164  5188  5212  5237  5261 

9 

5285  5310  5334  5358  5382  5406  5431  5455  5479  5503 

180 

25527  25551  25575  25600  25624  25648  25672  25696  25720  25744 

1 

5768  5792  5816  5840  5864  5888  5912  5935  5959  5983 

2 

6007  6031  6055  6079  6102  6126  6150  6174  6198  6221 

3 

6245  6269  6293  6316  6340  6364  6387  6411  6435  6458 

4 

6482  6505  6529  6553  6576  6600  6623  6647  6670  6(594 

5 

6717  6741  6764  6788  6811  6834  6858  6881  6905  6928 

6 

6951  6975  6998  7021  7045  7068  7091  7114  7138  7T(U 

7 

7184  7207  7231  7254  7277  7300  7323  7346  7370  7393 

8 

7416  7439  7462  7485  7508  7531  7554  7577  7600  7623 

9 

7646  7069  7692  7715  7738  7761  7784  7807  7830'  7852 

190 

27875  27898  27921  27944  27967  27989  28012  28035  28058  28081 

1 

8103  8126  8149  8171  8194  8217  8240  8262  8285  8307 

2 

8330  8353  8375  8398  8421  8443  8466  8488  8511  8533 

3 

8556  8578  8601  8623  8646  8668  8691  8713  8735  8758 

4 

8780  8803  8825  8847  8870  '8892  8914  8937  895p  8981 

5 

0003  9026  9048  9070  9092  9115  9137  9159  9181  9203 

6 

9226  9248  9270  9292  9314  9336  9358  9380  9403  9425 

7 

9447  9469  9491  9513  9535  9557  9579  9601  9623  9645 

8 

9667  9688  9710  t)732  9754  9776  9798  9820  9842  9863 

9 

9885  9907  9929  9951  9973  9994  30016  30038  30060  30081 

200 

30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 

00  o 

V  /^  fV 


TABLE  VI.— LOGARITHMS   OF   NUMBERS. 


N 

0123456789 

200 

30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 

1 

032»    0341    0363    0384    0406    0428    0449    0471    0492    0514 

2 

0535    0557    0578    0600    0621    0643    0664    0685    0707    0728 

3 

0750    0771    0792    0814    0835    0856    0878    0899    0920    0942 

4 

0963    0984    1006    1027    1048    1069    1091    1112    1133    1154 

5 

1175    1197    1218    1239    1260    1281    1302    1323    1345    1366 

6 

1387    1408    1429    1450    1471    1492    1513    1534    1555    1576 

7 

1597    1618    1639    1660    1681    1702    1723    1744    1765    1785 

8 

1806    1827    1848    1869    1890    1911    1931     1952    1973    1994 

9 

2015    2035    2056    2077    2098    2118    2139    2160    2181    2201 

210 

32222  32243  32263  32284  82305  32325  32346  32366  32387  32408 

1 

2428    2449    2469    2490    2510    2531    2552    2572    2593    2613 

2 

2634    2654    2675    2695    2715    2736    2756    2777    2797    2818 

3 

2838    2858    2879    2899    2919    2940    2960    2980    3001    3021 

4 

3041    3062    3082    3102    3122    3143    3163    3183    3203    3224 

5 

3244    3264    3284    3304    3325    3345    3365    3385    8405    3425 

6 

8445    3465    3486    3506    3526    8546    3566    8586    8606    3626 

7 

3646    3666    8686    3706    3726    8746    8766    8786    3806    8826 

8 

3846    3866    3885    3905    3925    3945    3965    3985    4005    4025 

9 

4044    4064    4084    4104    4124    4143    4163    4183    4203    4223 

220 

34242  34262  34282  34301  34321  84341  34361  34380  34400  84420 

1 

4439    4459    4479    4498    4518    4537    4557    4577    4596    4616 

2 

4635    4655    4674    4694    4713    4733    4753    4772    4792    4811 

3 

4830    4850    4869    4889    4908    4928    4947    4967    4986    5005 

4 

5025    5044    5064    5083    5102    5122    5141    5160    5180    5199 

5 

5218    5238    5257    5276    5295    5315    5334    5353    5372    5392 

6 

5411    5430    5449    5468    5488    5507    5526    5545    5564    5583 

7 

5603    5622    5641    5660    5679    5698    5717    5736    6755    5774 

8 

5793    5813    5832    5851    5870    5889    5908    5927    5946    5965 

9 

5984    6003    6021    6040    6059    6078    6097    6116    6135    6154 

230 

86173  36192  36211  86229  36248  36267  36286  36305  36324  36342 

1 

6361    6380    6399    6418    6436    6455    6474    6493    6511    6530 

2 

6549    6568    6586    6605    6624    6642    6661    6680    6698    6717 

3 

6736    6754    6773    6791    6810    6829    6847    6866    6884    6903 

4 

6922    6940    6959    6977    6996    7014    7033    7051    7070    7088 

5 

7107    7125    7144    7162    7181    7199    7218    7236    7254    7273 

6 

7291    7310    7328    7346    7365    7383    7401    7420    7438    7457 

7 

7475    7493    7511    7530    7548    7566    7585    7603    7621    7639 

8 

7658    7676    7694    7712    7731    7749    7767    7785    7803    7822 

9 

7840    7858    7876    7894    7912    7931    7949    7967    7985    8003 

240 

38021  38039  38057  38075  38093  38112  38130  38148  38166  38184 

1 

8202    8220    8238    8256    8274    8292    8310    8328    8346    8364 

2 

8382    8399    8417    8435    8453    8471    8489    8507    8525    8543 

3 

8561    8578    8596    8614    8632    8650    8668    8686    8703    8721 

4 

8739    8757    8775    8792    8810    8828    8846    8863    8881    8899 

5 

8917    8934    8952    8970    8987    9005    9023    9041    9058    9076 

6 

9094    9111    9129    9146    9164    9182    9199    9217    9235    9252 

7 

9270    9287    9305    9322    9340    9358    9375    9393    9410    9428 

8 

9445    9463    9480    9498    9515    9533    9550    9568    9585    9602 

9 

9620    9637    9655    9672    9690    9707    9724    9742    9759    9777 

250 

39794  39811  39829  39846  39863  39881  39898  39915  89933  39950 

TABLE  VI.— LOGARITHMS  OF   NUMBERS.  223 


N 

0123456789 

250 

39794  39811  39829  39846  39863  39881  39898  39915  39933  39950 

1 

9967  9985  40002  40019  40037  40054  40071  40088  40106  40123 

2 

40140  40157  0175  0192  0209  0226  0243  0261  0278  0295 

3 

0312  0329  0346  0364  0381  0398  0415  0432  0449  0466 

4 

0483  0500  0518;  0535  0552  0569  0586  0603  0620  0637 

5 

0654  0671  0688  0705  0722  0739  0756  0773  0790  0807 

6 

0824  0841  0858  0875  0892  0909  0926  0943  0960  097(5 

7 

0993  1010  1027  1044  1061  1078  1095  1111  1128  1145 

8 

1162  1179  1196  1212  1229  1246  1263  1280  1296  1313 

9 

1330  1347  1363  1380  1397  1414  1430  1447  1464  1481 

260 

41497  41514  41531  41547  41564  41581  41597  41614  41631  41647 

1 

1664  1681  1697  1714  1731  1747  1764  1780  1797  1814 

2 

1830  1847  1863  1880  1896  1913  1929  1946  1963  1979 

3 

1996  2012  2029  2045  2062  2078  2095  2111  2127  2144 

4 

2160  2177  2193  2210  2226  2243  2259  2275  2292  2308 

5 

2325  2341  2357  2374  2390  2406  2423  2439  2455  2472 

6 

2488  2504  2521  2537  2553  2570  2586  2602  2619  2635 

7 

2651  2667  2684  2700  2716  2732  2749  2765  2781  2797 

8 

2813  2830  2846  2862  2878  2894  2911  2927  2943  2959 

9 

2975  2991  3008  3024  3040  3056  3072  3088  3104  3120 

270 

43136  43152  43169  43185  43201  43217  43233  43249  43265  43281 

1 

3297  3313  3329  3345  3361  3377  3393  3409  3425  3441 

2 

3457  3473  3489  3505  3521  3537  3553  3569  3584  3600 

3 

3616  3632  3648  3664  3680  3696  3712  3727  3743  3759 

4 

3775  3791  3807  3823  3838  3854  3870  3886  3902  3917 

5 

3933  3949  3965  3981  3996  4012  4028  4044  4059  4075 

6 

4091  4107  4122  4138  4154  4170  4185  4201  4217  4232 

7 

4248  4264  4279  4295  4311  4326  4342  4358  4373  4389 

8 

4404  4420  4436  4451  4467  4483  4498  4514  4529  4545 

9 

4560  4576  4592  4607  4623  4638  4654  4669  4685  4700 

280 

44716  44731  44747  44762  44778  44793  44809  44824  44840  44855 

1 

4871  4886  4902  4917  4932  4948  4963  4979  4994  5010 

2 

5025  5040  5056  5071  5086  5102  5117  5133  5148  5163 

3 

5179  5194  5209  5225  5240  5255  5271  5286  5301  6317 

4 

5332  5347  5362  5378  5393  5408  5423  5439  5454  5469 

6 

5484  5500  5515  5530  5545  5561  5576  5591  5606  5621 

6 

5637  5652  5667  5682  5697  5712  5728  5743  5758  5773 

7 

5788  5803  5818  5834  5849  5864  5879  5894  5909  5924 

8 

5939  5954  5969  5984  6000  6015  6030  6045  6060  6075 

9 

6090  6105  6120  6135  6150  6165  6180  6195  6210  6225 

290 

46240  46255  46270  46285  46300  46315  46330  46345  46359  46374 

1 

6389  6404  6419  6434  6449  6464  6479  6494  6509  6523 

2 

6538  6553  6568  6583  6598  6613  6627  6642  6657  6672 

3 

6687  6702  6716  6731  6746  6761  6776  6790  6805  6820 

4 

6835  6850  6864  6879  6894  6909  6923  6938  6953  6967 

5 

6982  6997  7012  7026  7041  7056  7070  7085  7100  7114 

6 

7129  7144  7159  7173  7188  7202  7217  7232  7246  7261 

7 

7276  7290  7305  7319  7334  7349  7363  7378  7392  7407 

8 

7422  7436  7451  7465  7480  7494  7509  7524  7538  7553 

9 

7567  7582  7596  7611  7625  7640  7654  7669  7683  7698 

300 

47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 

i>0  1 

TABLE  VI.— liOGARITHMS  OF  NUMBEKS. 

N 
300 

Ol   2   3456789 

47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 

1 

7857  7871  7885  7900  7914  7929  7943  7958  7972  7986 

2 

8001  8015  8029  8044  8058  8073  8087  8101  8116  8130 

3 

8144  8159  8173  8187  8202  8216  8230  8244  8259  8273 

4 

8287  8302  8316  8330  8344  8359  8373  8387  8401  8416 

5 

8430  8444  8458  8473  8487  8501  8515  8530  8544  8558 

6 

8572  8586  8601  8615  8629  8643  8657  8671  8686  8700 

7 

8714  8728  8742  8756  8770  8785  8799  8813  8827  8841 

8 

8855  8869  8883  8897  8911  8926  8940  8954  8968  8982 

9 

8996  9010  9024  9038  9052  9066  9080  9094  9108  9122 

310 

49136  49150  49164  49178  49192  49206  49220  49234  49248  49262 

1 

9276  9290  9304  9318  9332  9346  9360  9374  9388  9402 

2 

9415  9429  9443  9457  9471  9485  9499  9513  9527  9541 

3 

9554  9568  9582  9596  9610  9624  9638  9651  9665  9679 

4 

9693  9707  9721  9734  9748  9762"  9776  9790  9803  9817 

5 

9831  9845  9859  9872  9886  9900  9914  9927  9941  9955 

6 

9969  9982  9996  50010  50024  50037  50051  60065  50079  50092 

7 

50106  50120  50133  0147  0161  0174  0188  0202  0215  0229 

8 

0243  0256  0270  0284  0297  0311  0325  0338  0362  0365 

9 

0379  0393  0406  O420  0433  0447  046l  0474  0488  0501 

320 

50515  60629  60542  50556  60660  60683  50596  50610  60623  50637 

1 

0661  0664  0678  '0691  0705  0718  0732  0745  0769  0772 

2 

0786  0799  0813  0826  0840  0853  0866  0880  0893  0907 

3 

0920  0934  0947  0961  0974  0987  1001  1014  1028  1041 

4 

1065  1068  1081  1095  1108  1121  1135  1148  1162  1175 

5 

1188  1202  1215  1228  1242  1255  1268  1282  1295  1308 

6 

1322  1335  1348  1362  1375  1388  1402  1415  1428  1441 

7 

1455  1468  1481  1495  1508  1621  1634  1648  1661  1574 

8 

1687  1601  1614  1627  1640  1654  1667  1680  1693  1706 

9 

1720  1733  1746  1759  1772  1786  1799  1812  1825  1838 

330 

61851  61865  51878  61891  61904  61917  61930  51943  51957  51970 

1 

1983  1996  2009  2022  2035  2048  2061  207i  2088  2101 

2 

2114  2127  2140  2153  2166  2179  2192  ,2206  2218  2231 

3 

2244  2257  2270  2284  2297  2310  2323  233^  2349  2362 

4 

2375  2388  2401  2414  2427  2440  2453  2466  2479  2492 

5 

2604  2517  2630  2643  2556  2669  2582  2595-  2608  2621 

6 

2634  2647  2660  2673  2686  2699  2711  2724  2737  2750 

7 

2763  2776  2789  2802  2815  2827  2840  2853  2866  2879 

8 

2892  2905  2917  2930  2943  2956  2969  2982  2994  3007 

9 

3020  3033  3046  3058  3071  3084  3097  3110  3122  3136 

340 

63148  53161  63173  53186  63199  63212  63224  53237  63250  53263 

1 

3275  3288  3301  3314  3326  3339  3352  3364  3377  3390 

2 

3403  3415  3428  3441  3453  3466  3479  3491  3504  3617 

3 

3529  3542  3555  3567  3580  3593  3605  3618  3631  ^.643 

4 

3656  3668  3681  3694  3706  3719  3732  3744  3757  21769 

5 

3782  3794  3807  3820  3832  3845  3857  3870  3882  ;-1895 

6 

3908  3920  3933  3945  3958  3970  3983  3995  4008  4  020 

7 

4033  4045  4058  4070  4083  4095  410^  4120  4133  4145 

8 

4158  4170  4183  4195  4208  4220  4233  4245  4258  4270 

9 

4283  4295  4307  4320  4332  4345  4357  4370.  4382  4394 

350 

54407  54419  64432  54444  54466  54469  64481  64494  54606  64(  ')18 

TABLE  VI.— LOGARITHMS  OF   NUMBERS.  22.J 


N 


350 
1 
2 
3 
4 
5 
6 
7 
8 
9 

360 
1 
2 
3 
4 
5 
6 
7 
8 
9 

370 

1 
2 
3 

4 
5 
6 
7 
8 
9 

380 

1 
2 
3 
4 
5 
6 
7 
8 
9 

390 

1 
2 
3 

4 
5 
6 
7 
8 


0    12   34   567   8   9 


54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 

4531  4543  4555  4568  4580  4593  4605  4617  4630  4642 

4654  4667  4679  4691  4704  4716  4728  4741  4753  4765 

4777  4790  4802  4814  4827  4839  4851  4864  4876  4888 

4900  4913  4925  4937  4949  4962  4974  4986  4998  5011 

5023  5035  5047  5060  5072  5084  5096  5108  5121  5133 

5145  5157  5169  5182  5194  5206  5218  5230  5242  5255 

5267  5279  5291  5303  5315  5328  5340  5352  5364  5376 

5388  5400  5413  5425  5437  5449  5461  5473  5485  5497 

5509  5522  5534  5546  5558  5570  5582  5594  5606  5618 

55630  55642  55654  55666  55678  55691  55703  55715  55727  55739 

5751  5763  5775  .5787  5799  5811  5823  5835  5847  5859 

5871  5883  5895  5907  5919  5931  5943  5955  5967  5979 

5991  6003  6015  6027  6038  6050  6062  6074  6086  6098 

6110  6122  6134  6146  6158  6170  6182  6194  6205  6217 

6229  6241  6253  6265  6277  6289  6301  6812  6324  6336 

6348  6360  6372  6384  6396  6407  6419  6431  6443  6455 

6467  6478  6490  6502  6514  6526  6538  6549  6561  6573 

6585  6597  6608  6620  6632  6644  6656  6667  6679  6691 

6703  6714  6726  6738  6750  6761  6773  6785  6797  6808 

56820  56832  56844  56855  56867  56879  56891  56902  56914  56926 

6937  6949  6961  6972  6984  6996  7008  7019  7031  7043 

7054  7066  7078  7089  7101  7113  7124  7136  7148  7159 

7171  7183  7194  7206  7217  7229  7241  7252  7264  7276 

7287  7299  7310  7322  7334  7345  7357  7368  7380  7392 

7403  7415  7426  7438  7449  7461  7473  7484  7496  7507 

7519  7530  7542  7553  7565  7576  7588  7600  7611  7623 

7634  7646  7657  7669  7680  7692  7703  7715  7726  7738 

7749  7761  7772  7784  7795  7807  7818  7830  7841  7852 

7864  7875  7887  7898  7910  7921  7933  7944  7955  7967 

57978  57990  58001  58013  58024  58035  58047  58058  58070  58081 

8092  8104  8115  8127  8138  8149  8161  8172  8184  8195 

8206  8218  8229  8240  8252  8263  8274  8286  8297  8309 

8320  8331  8343  8354  8365  8377  8388  8399  8410  8422 

8433  8444  8456  8467  8478  8490  8501  8512  8524  8535 

8546  8557  8569  8580  8591  8602  8614  8625  8636  8647 

8659  8670  8681  8692  8704  8715  8726  8737  8749  8760 

8771  8782  8794  8805  8816  8827  8838  8850  8861  8872 

8863  8894  8906  8917  8928  8939  8950  8961  8973  8984 

8995  9006  9017  9028  9040  9051  9062  9073  9084  9095 

59106  59118  59129  59140  59151  59162  59173  59184  59195  59207 

9218  9229  9240  9251  9262  9273  9284  9295  9306  9318 

9329  9340  9351  9362  9373  9384  9395  9406  9417  9428 

9439  9450  9461  9472  9483  9494  9506  9517  9528  9539 

9550  9561  9572  9583  9594  9605  9616  9627  9638  9649 

9660  9671  9682  9693  9704  9715  9726  9737  9748  9759 

9770  9780  9791  9802  9813  9824  9835  9846  9857  9868 

9879  9890  9901  9912  9923  9934  9945  9956  9966  9977 

9988  9999  60010  60021  60032  60043  60054  60065  60076  60086 


9  60097  60108  0119  0130  0141  0152  0103  0173  0184  0195 
400  60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 


226  TABLE  VI.— LOGARITHMS   OF  NUMBERS. 


N 
400 

O   1   3   3 

4   5   6   7 

8   9 

60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 

1 

0314  0325  0336  0347 

0358  0369  0379  0390 

0401  0412 

2 

0423  0433  0444  0455 

0466  0477  0487  0498 

0509  0520 

3 

0531  0541  0552  0563 

0574  0584  0595  0606 

0617  0627 

4 

0638  0649  0660  0670 

0681  0692  0703  0713 

0724  0735 

5 

0746  0756  0767  0778 

0788  0799  0810  0821 

0831  0842 

6, 

0853  0863  0874  0885 

0895  0906  0917  0927 

0938  0949 

7 

0959  0970  0981  0991 

1002  1013  1023  1034 

1045  1055 

8 

1066  1077  1087  1098 

1109  1119  1130  1140 

1151  1162 

9 

1172  1183  1194  1204 

1215  1225  1236  1247 

1257  1268 

410 

61278  61289  61300  61310  61321  61331  61342  61352 

61363  61374 

1 

1384  1395  1405  1416 

1426  1437  1448  1458 

1469  1479 

2 

1490  1500  1511  1521 

1532  1542  1553  1563 

1574  1584 

3 

1595  1606  1616  1627 

1637  1648  1658  1669 

1679  1690 

4 

1700  1711  1721  1731 

1742  1752  1763  1773 

1784  1794 

5 

1805  1815  1826  1836 

1847  1857  1868  1878 

1888  1899 

6 

1909  1920  1930  1941 

1951  1962  1972  1982 

1993  2003 

7 

2014  2024  2034  2045 

2055  2066  2076  2086 

2097  2107 

8 

2118  2128  2138  2149 

2159  2170  2180  2190 

2201  2211 

9 

2221  2232  2242  2252 

2263  2273  2284  2294 

2304  2315 

420 

62325  62335  62346  62356  62366  62377  62387  62397  62408  62418 

1 

2428  2439  2449  2459 

2469  2480  2490  2500 

2511  2521 

2 

2531  2542  2552  2562 

2572  2583  2593  2603 

2613  2624 

3 

2634  2644  2655  2665 

2675  2685  2696  2706 

2716  2726 

4 

2737  2747  2757  2767 

2778  2788  2798  2808 

2818  2829 

5 

2839  2849  2859  2870 

2880  2890  2900  2910 

2921  2931 

6 

2941  2951  2961  2972 

2982  2992  3002  3012 

3022  3033 

7 

3043  3053  3063  3073 

3083  3094  3104  3114 

3124  3134 

8 

3144  3155  3165  3175 

3185  3195  3205  3215 

3225  3236 

9 

3246  3256  3266  3276 

3286  3296  3306  3317 

3327  3337 

430 

63347  63357  63367  63377 

63387  63397  63407  63417  63428  63438 

1 

3448  3458  3468  3478 

3488  3498  3508  3518 

3528  3538 

2 

3548  3558  3568  3579 

3589  3599  3609  3619 

3629  3639 

3 

3649  3659  3669  '  3679 

3689  3699  3709  3719 

3729  3739 

4 

3749  3759  3769  3779 

3789  3799  3809  3819 

3829  3839 

5 

3849  3859  3869  3879 

3889  3899  3909  3919 

3929  3939 

6 

3949  3959  3969  3979 

3988  3998  4008  4018 

4028  4038 

7 

4048  4058  4068  4078 

4088  4098  4108  4118 

4128  4137 

8 

4147  4157  4167  4177 

4187  4197  4207  4217 

4227  4237 

9 

4246  4256  4266  4276 

4286  4296  4306  4316 

4326  4835 

440 

64345  64355  64365  64375  64385  64395  64404  64414  64424  64434 

1 

4444  4454  4464  4473 

4483  4493  4503  4513 

4523  4532 

2 

4542  4552  4562  4572 

4582  4591  4601  4611 

4621  4631 

3 

4640  4650  4660  4670 

4680  4689  4699  4709 

4719  4729 

4 

4738  4748  4758  4768 

4777  4787  4797  4807 

4816  4826 

5 

4836  4846  4856  4865 

4875  4885  4895  4904 

4914  4924 

6 

4933  4943  4953  4963 

4972  4982  4992  5002 

5011  5021 

7 

5031  5040  5050  5060 

5070  5079  5089  5099 

5108  5118 

8 

5128  5137  5147  5157 

5167  5176  5186  5196 

5205  5215 

9 

5225  5234  5244  5254 

5263  5273  5283  5292 

5302  5312 

450 

65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 

r^ 


TABLE  VI.— J.O(JAHITIIMS   OF   NTMBEHS.  227 


N 

O       1        2        3 

4        5       6        7        8        9 

450 

65321  05831  65341  65350  65360  65360  65370  65380  65308  65408 

1 

5418    5427    5437    5447 

5456    5466    5475    5485    5405    5504 

2 

5514    5523    5533    5543 

5552    5562    5571    5581    5501    5600 

3 

5610    5610    5620    5630 

5648    6658    5667    5677    5686    5606 

4 

5706    5715    5725    5734 

5744    5753    5763    5772    5782    5702 

5 

5801    5811    5820    5830 

5830    5840    5858    5868    5877    5887 

6 

5806    5006    5016    5025 

5035    5044    5054    5063    5973    5082 

7 

5002    6001    6011    6020 

6030    6030    6040    6058    6068    6077 

8 

6087    6006    6106    6115 

6124    6134    6143    6153    6162    6172 

9 

6181    6101    6200    6210 

6210    6220    6238    6247    6257    6266 

460 

66276  66285  66205  66304  66314  66323  66332  66342  66351  66361 

1 

6370    6380    6380    6308 

6408    6417    6427    6436    6445    6455 

2 

6464    6474    6483    6402 

6502    6511    6521    6580    6530    6540 

3 

6558    6567    6577    6586 

6506    6605    6614    6624    6633    6642 

4 

6652    6661    6671    6680 

6680    6600    6708    6717    6727    6736 

5 

6745    6755    6764    6773 

6783    6702    6801    6811    6820    6820 

-§ 

6830    6848    6857    6867 

6876    6885    6804    6004    6013    6022 

7 

6032    6041    6050    6060 

6969    697 S    6087    6007    7006    7015 

S 

7025    7034    7043    7052 

7062    7071    7080    7080    7000    7108 

9 

7il7    7127    7136    7145 

7154    7164    7173    7182    7101    7201 

470 

67210  67210  67228  67237  67247  67256  67265  67274  67284  67203 

1 

7302    7311    7321    7330 

7380    7848    7857    7867    7376    7885 

2 

7304    7403    7413    7422 

7481    7440    7440    7450    7468    7477 

3 

7486    7405    7504    7514 

7523    7532    7541    7550    7560    7560 

4 

7578    7587    7506    7605 

7614    7624    7638    7642    7651    7660 

5 

7660    7670    7688    7607 

7706    7715    7724    7733    7742    7752 

6 

7761    7770    7770    7788 

7707    7806    7815    7825    7834    7843 

7 

7852    7861    7870    7870 

7888    7807    7006    7016    7025    7034 

8 

7043    7052    7061    7070 

7070    7088    7007    8006    8015    8024 

9 

8034    8043    8052    8061 

8070    8070    8088    8007    8106    8115 

480 

68124  68133  68142  68151 

68160  68160  68178  68187  68196  68205 

1 

8215    8224    -8233    8242 

8251    8260    8260    8278    8287    8206 

2 

8305    8314    8323    «332 

8341    8350    8350    8368    8377    8386 

3 

8305    8404    8413    8422 

8431    8440    8440    8458    8467    8476 

4 

8485    8404    8502    8511 

8520    8520    8588    8547    8556    8565 

5 

8574    8583    8502    8601 

8610    8610    8628    8637    8646    8655 

6 

8664    8673    8681    8600 

8600    8708    8717    8726    8785    8744 

7 

8753    8762    8771    8780 

8780    8707    8806    8815    8824    8833 

8 

8842    8851    8860    8860 

8878    8886    8805    8004    8018    8022 

9 

8031    8040    8040    8058 

8066    8075    8084    8003    0002    0011 

490 

60020  60028  60037  60046  60055  60064  60073  60082  60000  60000 

1 

0108    0117    0126    0135 

0144    0152    0161    0170    0170    0188 

2 

0107    0205    0214    0223 

0232    0241    0240    0258    0267    0276 

3 

9285    0294    0302    9311 

0320    0320    0338    0346    0355    9364 

4 

0373    0381    0300    0300 

9408    9417    9425    9434    9443    9452 

5 

0461    0460    0478    0487 

9406    0504    0513    0522    0531    0530 

6 

0548    0557    0566    0574 

0583    0502    0601    9600    0618    0627 

7 

0636    0644    0653    9662 

0671    0670    0688    0607    0705    0714 

8 

0723    0732    0740    0740 

0758    0767    0775    0784    0703    0801 

9 

0810    0810    0827    0836 

0845    0854    0862    0871    0880    0888 

500 

60807  60006  60014  60023 

60032  60040  60040  60058  00066  60075 

228  TABLE  VI —LOGARITHMS   OF    XTMBERS. 


N 

O 

1    2 

3 

4    5 

6 

7 

8 

9 

500 

69897  69906  69914  69923  69932  69940  69949  69958  69966  69975  | 

1 

9984 

9992  70001  70010 

70018  70027  70036  70044 

70053 

70062 

2 

70070  70079  0088 

0096 

0105  0114 

0122 

0131 

0140 

0148 

3 

0157 

0165  0174 

0183 

0191  0200 

0209 

0217 

0226 

0234 

4 

0243 

0252  0260 

0269 

0278  0286 

0295 

0303 

0312 

0321 

5 

0329 

0338  0346 

0355 

0364  0372 

0381 

0389 

0398 

0406 

6 

0415 

0424  0432 

0441 

0449  0458 

0467 

0475 

0484 

0492 

7 

0501 

0509  0518 

0526 

0535  0544 

0552 

0561 

0569 

0578 

8 

0586 

0595  0603 

0612 

0621  0629 

0638 

0646 

0655 

0663 

9 

0672 

0680  0689 

0697 

0706  0714 

0723 

0731 

0740 

0749 

510 

70757 

70766  70774 

70783 

70791  70800  70808 

70817 

70825 

70834 

1 

0842 

0851  0859 

0868 

0876  0885 

0893 

0902 

0910 

0919 

2 

0927 

0935  0944 

0952 

0961  0969 

0978 

0986 

0995 

1003 

3 

1012 

1020  1029 

1037 

1046  1054 

1063 

1071 

1079 

1088 

4 

1096 

1105  1113 

1122 

1130  1139 

1147 

1155 

1164 

1172 

5 

1181 

1189  1198 

1206 

1214  ■  1223 

1231 

1240 

1248 

1257 

6 

1265 

1273  1282 

1290 

1299  1307 

1315 

1324 

1332 

1341 

7 

1349 

1357  1366 

1374 

1383  1391 

1399 

1408 

1416 

1425 

8 

1433 

1441  1450 

1458 

1466  1475 

1483 

1492 

1500 

1508 

9 

1517 

1525  1533 

1542 

1550  1559 

1567 

1575 

1584 

1592 

520 

71600 

71609  71617  71625  71634  71642 

71650  71659  71667  71675 

1 

1684 

1692  1700 

1709 

1717  1725 

1734 

1742 

1750 

1759 

2 

1767 

1775  1784 

1792 

1800  1809 

1817 

1825 

1834 

1842 

3 

1850 

1858  1867 

1875 

1883  1892 

1900 

1908 

1917 

1925 

4 

1933 

1941  1950 

1958 

1966  1975 

1983 

1991 

1999 

2008 

5 

2016 

2024  2032 

2041 

2049  2057 

2066 

2074 

2082 

2090 

6 

2099 

2107  2115 

2123 

2132  2140 

2148 

2156 

2165 

2173 

7 

2181 

2189  2198 

2206 

2214  2222 

2230 

2239 

2247 

2255 

8 

2263 

2272  2280 

2288 

2296  2304 

2313 

2321 

2329 

2337 

9 

2346 

2354  2362 

2370 

2378  2387 

2395 

2403 

2411 

2419 

530 

72428 

72436  72444  72452  72460  72469  72477 

72485  72493  72501 

1 

2509 

2518  2526 

2534 

2542  2550 

2558 

2567 

2575 

2583 

2 

2591 

2599  2607 

2616 

2624  26.^ 

2640 

2648 

2656 

2665 

3 

2673 

2681  2689 

2697 

2705  2713 

2722 

2730 

2738 

2746 

4 

2754 

2762  2770 

2779 

2787  2795 

2803 

2811 

2819 

2827 

5 

2835 

2843  2852 

2860 

2868  2876 

2884 

2892 

2900 

2908 

6 

2916 

2925  2933 

2941 

2949  2957 

2965 

2973 

2981 

2989 

7 

2997 

3006  3014 

3022 

3030  3038 

3046 

3054 

3062 

3070 

8 

3078 

3086  3094 

3102 

3111  3119 

3127 

3135 

3143 

3151 

9 

3159 

3167  3175 

3183 

3191  3199 

3207 

3215 

3223 

3231 

540 

73239  73247  73255 

73263  73272  73280  73288  73296  73304  73312 

1 

3320 

3328  3336 

3344 

3352  3360 

3368 

3376 

3384 

3392 

2 

3400 

3408  3416 

3424 

3432  3440 

3448 

3456 

3464 

3472 

3 

3480 

3488  3496 

3504 

3512  3520 

3528 

3536 

3544 

3552 

4 

3560 

3568  3576 

3584 

3592  3600 

3608 

3616 

3624 

3632 

5 

3640 

3648  3656 

3664 

3672  3679 

3687 

3695 

3703 

3711 

6 

3719 

3727  3735 

3743 

3751  3759 

3767 

3775 

3783 

3791 

7 

3799 

3807  3815 

3823 

3830  3838 

3846 

3854 

3862 

3870 

8 

3878 

3886  3894 

3902 

3910  3918 

3926 

3933 

3941 

3949 

9 

3957 

3965  3973 

3981 

3989  3997 

4005 

4013 

4020 

4028 

550 

74036  74044  74052  74060  74068  74076  74084 

74092 

74099 

74107 
1 

TABLE  VI.— LOGARITHMS   OF   NUMBERS. 


229 


N 

O    1    2    3   4    5   6 

7    8   9 

650 

74036  74044  74052  74060  74068  74076  74084  74002  74099  74107 

1 

4115  4128  4131  4139  4147  4155  4162 

4170  4178  4186 

2 

4194  4202  4210  4218  4225  4233  4241 

4249  4257  4265 

3 

4273  4280  4288  4296  4304  4312  4320 

4327  4335  4343 

4 

4351  4359  4367  4374  4382  4390  4398 

4406  4414  4421 

5 

4429  4437  4445  4453  4461  4468  4476 

4484  4492  4500 

6 

4507  4515  4523  4531  4539  4547  4554 

4562  4570  4578 

7 

4586  4593  4601  4609  4617  4624  4632 

4640  4648  4656 

8 

4663  4671  4679  4687  4695  4702  4710 

4718  4726  4733 

9 

4741  4749  4757  4764  4772  4780  4788 

4796  4803  4811 

560 

74819  74827  74834  74842  74850  74858  74865 

74873  74881  74889 

1 

4896  4904  4912  4920  4927  4935  4943 

4950  4958  4966 

2 

4974  4981  4989  4997  5005  5012  5020 

5028  5035  5043 

3 

5051  5059  5066  5074  5082  5089  5097 

5105  5113  5120 

4 

5128  5136  5143  5151  5159  5166  5174 

5182  5189  5197 

5 

5205  5213  5220  5228  5236  5243  5251 

5259  5266  5274 

6 

5282  5289  5297  5305  5312  5320  5328 

5335  5343  5351 

7 

5358  5366  5374  5381  5389  5397  5404 

5412  5420  5427 

8 

5435  5442  5450  5458  5465  5473  5481 

5488  5496  5504 

9 

5511  5519  5526  5534  5542  5549  5557 

5565  5572  5580 

570 

75587  75595  75603  75610  75618  75626  75633  75641  75648  75656 

1 

5664  5671  5679  5686  5694  5702  5709 

5717  5724  5732 

2 

5740  5747  5755  5762  5770  5778  5785 

5793  5800  5808 

3 

5815  5823  5831  5838  5846  5853  5861 

5868  5876  5884 

4 

5891  5899  5906  5914  5921  5929  5937 

5944  5952  5959 

5 

5967  5974  5982  5989  5997  6005  6012 

6020  6027  6035 

6 

6042  6050  6057  6065  6072  6080  6087 

6095  6103  6110 

7 

6118  6125  6133  6140  6148  6155  6163 

6170  6178  6185 

8 

6193  6200  0208  6215  6223  6230  6238 

6245  6253  6260 

9 

6268  6275  6283  6290  6298  6305  6313 

6320  6328  6335 

580 

76343  76350  76358  76365  76373  76380  76388  76395  76403  76410 

1 

6418  6425  6433  6440  6448  6455  6462 

6470  6477  6485 

2 

6492  6500  6507  6515  6522  6530  6537 

6545  6552  6559 

3 

6567  6574  6582  6589  6597  6604  6612 

6619  6626  6634 

4 

6641  6649  6656  6664  6671  6678  6686 

6693  6701  6708 

5 

6716  6723  6730  6738  6745  6753  6760 

6768  6775  6782 

6 

6790  6797  6805  6812  6819  6827  6834 

6842  6849  6856 

7 

6864  6871  6879  6886  6893  6901  6908 

6916  6923  6930 

8 

0938  694^  6953  6960  6967  6975  6982 

6989  6997  7004 

9 

7012  7019  7026  7034  7041  7048  7056 

7063  7070  7078 

590 

77085  77093  77100  77107  77115  77122  77129  77137  77144  77151 

1 

7159  7166  7173  7181  7188  7195  7203 

7210  7217  7225 

2 

7232  7240  7247  7254  7262  7269  7276 

7283'  7291  7298 

3 

7305  7313  7320  7327  7335  7342  7349 

7357  7364  7371 

4 

7379  7386  7393  7401  7408  7415  7422 

7430  7437  7444 

5 

7452  7459  7466  7474  7481  7488  7495 

7503  7510  7517 

6 

7525  7532  7539  7546  7554  7561  7568 

7576  7583  7590 

7 

7597  7605  7612  7619  7627  7634  7641 

7648  7656  7663 

8 

7670  7677  7685  7692  7699  7706  7714 

7721  7728  7735 

9 

7743  7750  7757  7764  7772  7779  7786 

7793  7801  7808. 

600 

77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 

TABLE  VI.— LOGARITHMS   OF  NUMBERS. 


N 

O   1   2   3   4   5 

6 

7    8   9 

600 

77815  77822  77830  77837  77844  77851 

77859  77866  77873  77880 

1 

7887  7895  7902  7909  7916  7924 

7931 

7938  7945  7952 

2 

7960  7967  7974  7981  7988  7996 

8003 

8010  8017  8025 

3 

8032  8039  8046  8053  8061  8068 

8075 

8082  8089  8097 

4 

8104  8111  8118  8125  8132  8140 

8147 

8154  8161  8168 

5 

8176  8183  8190  8197  8204  8211 

8219 

8226  8233  8240 

6 

8247  8254  8262  8269  8276  8283 

8290 

8297  8305  8312 

7 

8319  8326  8333  8340  8347  8355 

8362 

8369  8376  8383 

8 

8390  8398  8405  8412  8419  8426 

8433 

8440  8447  8455 

9 

8462  8469  8476  8483  8490  8497 

8504 

8512  8519  8526 

610 

78533  78540  78547  78554  78561  78569  78576  78583  78590  78597 

1 

8604  8611  8618  8625  8633  8640 

8647 

8654  8661  8668 

2 

8675  8682  8689  8696  8704  8711 

8718 

8725  8732  8739 

3 

8746  8753  8760  8767  8774  8781 

8789 

8796  8803  8810 

4 

8817  8824  8831  8838  8845  8852 

8859 

8866  8873  8880 

5 

8888  8895  8902  8909  8916  8923 

8930 

8937  8944  8951 

6 

8958  8965  8972  8979  8986  8993 

9000 

9007  9014  9021 

7 

9029  9036  9043  9050  9057  9064 

9071 

9078  9085  9092 

8 

9099  9106  9113  9120  9127  9134 

9141 

9148  9155  9162 

9 

9169  9176  9183  9190  9197  9204 

9211 

9218  9225  9232 

620 

79239  79246  79253  79260  79267  79274 

79281 

79288  79295  79302 

1 

9309  9316  9323  9330  9337  9344 

9351 

9358  9365  9372 

2 

9379  9386  9393  9400  9407  9414 

9421 

9428  9435  9442 

3 

9449  9456  9463  9470  9477  9484 

9491 

9498  9505  9511 

4 

9518  9525  9532  9539  9546  9553 

9560 

9567  9574  9581 

5 

9588  9595  9602  9609  9616  9623 

9630 

9637  9644  9650 

6 

9657  9664  9671  9678  9685  9692 

9699 

9706  9713  9720 

7 

9727  9734  9741  9748  9754  9761 

9768 

9775  9782  9789 

8 

9796  9803  9810  9817  9824  9831 

9837 

9844  9851  9858 

9 

9865  9872  9879  9886  9893  9900 

9906 

9913  9920  9927 

630 

79934  79941  79948  79955  79962  79969  79975 

79982  79989  79996 

1 

80003  80010  80017  80024  80030  80037  80044  80051  80058  80065  | 

2 

0072  0079  0085  0092  0099  0106 

0113 

0120  0127  0134 

3 

0140  0147  0154  0161  0168  0175 

0182 

0188  0195  0202 

4 

0209  0216  0223  0229  0236  0243 

0250 

0257  0264  0271 

5 

0277  0284  0291  0298  0305  0312 

0318 

0325  0332  0339 

6 

0346  0353  0359  0366  0373  0380 

0387 

0393  0400  0407 

7 

0414  0421  0428  0434  0441  0448 

0455 

0462  0468  0475 

8 

0482  0489  0496  0502  0509  0516 

0523 

0530  0536  0543 

9 

0550  0557  0564  0570  0577  0584 

0591 

0598  0604  0611 

640 

80618  80625  80632  80638  80645  80652  80659  80665  80672  80679  | 

1 

0686  0693  0699  0706  0713  0720 

0726 

0733  0740  0747 

2 

0754  0760  0767  0774  0781  0787 

0794 

0801  0808  0814 

3 

0821  0828  0835  0841  0848  0855 

0862 

0868  0875  0882 

4 

0889  0895  0902  0909  0916  0922 

0929 

0936  0943  0949 

5 

0956  0963  0969  0976  0983  0990 

0996 

1003  1010  1017 

3 

1023  1030  1037  1043  1050  1057 

1064 

1070  1077  1084 

7 

1090  1097  1104  1111  1117  1124 

1131 

1137  1144  1151 

8 

1158  1164  1171  1178  1184  1191 

1198 

1204  1211  1218 

9 

1224  1231  1238  1245  1251  1258 

1265 

1271  1278  1285 

650 

1 

81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 

TABLE  VI.— LOGARITHMS  OF   NUMBEKB. 


331 


N 

01234567    89 

650 

81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 

1 

1358  1365  1371  1378  1385  1391  1398  1405  1411  1418 

2 

1425  1431  1438  1445  1451  1458  1465  1471  1478  1485 

3 

1491  1498  1505  1511  1518  1525  1531  1538  1544  1551 

4 

1558  1564  1571  1578  1584  1591  1598  1604  1611  1617 

5 

1624  1631  1637  1644  1651  1657  1664  1671  1677  1684 

6 

1690  1697  1704  1710  1717  1723  1730  1737  1743  1750 

7 

1757  1763  1770  1776  1783  1790  1796  1803  1809  1816 

8 

1823  1829  1836  1842  1849  1856  1862  1869  1875  1882 

9 

1889  1895  1902  1908  1915  1921  1928  1935  1941  1948 

660 

81954  81961  81968  81974  81981  81987  81994  82000  82007  82014 

1 

2020  2027  2033  2040  2046  2053  2060  2066  2073  2079 

2 

2086  2092  2099  2105  2112  2119  2125  2132  2138  2145 

3 

2151  2158  2164  2171  2178  2184  2191  2197  2204  2210 

4 

2217  2223  2230  2236  2243  2249  2256  2263  2269  2276 

5 

2282  2289  2295  2302  2308  2315  2321  2328  2334  2341 

6 

2347  2354  2360  2367  2373  2380  2387  2393  2400  2406 

7 

2413  2419  2426  2432  2439  2445  2452  2458  2465  2471 

8 

2478  2484  2491  2497  2504  2510  2517  2523  2530  2536 

9 

2543  2549  2556  2562  2569  2575  2582  2588  2595  2601 

670 

82607  82614  82620  82627  82633  82640  82646  82653  82659  82666 

1 

2672  2679  2685  2692  2698  2705  2711  2718  2724  2730 

2 

2737  2743  2750  2756  2763  2769  2776  2782  2789  2795 

3 

2802  2808  2814  2821  2827  2834  2840  2847  2853  2860 

4 

2866  2872  2879  2885  2892  2898  2905  2911  2918  2924 

5 

2930  2937  2943  2950  2956  2963  2969  2975  2982  2988 

6 

2995  3001  3008  3014  3020  3027  3033  3040  3046  3052 

7 

3059  3065  3072  3078  3085  3091  3097  3104  3110  3117 

8 

3123  3129  3136  3142  3149  315i  3161  3168  3174  3181 

9 

3187  3193  3200  3206  3213  3219  3225  3232  3238  3245 

680 

83251  83257  83264  83270  83276  83283  83289  83296  83302  83308 

1 

3315  3321  3327  3334  3340  3347  3353  3359  3366  3372 

2 

3378  3385  3391  3398  3404  3410  3417  3423  3429  3436 

3 

3442  3448  3455  3461  3467  3474  3480  3487  3493  3499 

4 

3506  3512  3518  3525  3531  3537  3544  3550  3556  3563 

5 

3569  3575  3582  3588  3594  3601  3607  3613  3620  3626 

6 

3632  3639  3645  3651  3658  3664  3670  3677  3683  3689 

7 

3696  3702  3708  3715  3721  3727  3734  3740  3746  3753 

8 

3759  3765  3771  3778  3784  3790  3797  3803  3809  3816 

9 

3822  3828  3835  3841  3847  3853  3860  3866  3872  3879 

690 

83885  83891  83897  83904  83910  83916  83923  83929  83935  83942 

1 

3948  3954  3960  3967  3973  3979  3985  3992  3998  4004 

2 

4011  4017  4023  4029  4036  4042  4048  4055  4061  4067 

3 

4073  4080  4086  4092  4098  4105  4111  4117  4123  4130 

4 

4136  4142  4148  4155  4161  4167  4173  4180  4186  4192 

5 

4198  4205  4211  4217  4223  4230  4236  4242  4248  4255 

6 

4261  4267  4273  4280  4286  4292  4298  4305  4311  4317 

7 

4323  4330  4336  4342  4348  4354  4361  4367  4373  4379 

8 

4386  4392  4398  4404  4410  4417  4423  4429  4435  4442 

9 

4448  4454  4460  4466  4473  4479  4485  4491  4497  4504 

700 

84510  84516  84522  84528  84535  84541  84547  84553  84559  84566 

/^    >J   ryj 


TABLE  VI.— LOGARITHMS  OF   NUMBERS. 


N 


0123456789 


700  '  84510  84516  84522  84528  84535  84541  84547  84553  84559  84566 

1  !    4572  4578  4584  4590  4597  4603  4609  4615  4621  4628 

2  j    4634  4640  4646  4652  4658  4665  4671  4677  4683  4689 

3  I    4696  4702  4708  4714  4720  4726  4733  4739  4745  4751 

4  i    4757  4763  4770  4776  4782  4788  4794  4800  4807  4813 

5  4819  4825  4831  4837  4844  4850  4856  4862  4868  4874 

6  4880  4887  4893  4899  4905  4911  4917  4924  4930  4936 

7  4942  4948  4954  4960  4967  4973  4979  4985  4991  4997 

8  5003  5009  5016  5022  5028  5034  5040  5046  5052  5058 

9  5065  5071  5077  5083  5089  5095  5101  5107  5114  5120 


710 

1 
2 
3 
4 
5 
6 
7 
8 
9 


85126  85132  85138  85144  85150  85156  85163  85169  85175  85181 

5187  5193  5199    5205  5211  5217  5224  5230  5236  5242 

5248  5254  5260  5266  5272  5278  5285  5291  5297  5303 

5309  5315  5321  5327  5333  5339  5345  5352  5358  5364 

5370  5376  5382  5388  5394  5400  5406  5412  5418  5425 

5431  5437  5443  5449  5455  5461  5467  5473  5479  5485 

5491  5497  5503  5509  5516  5522  5528  5534  5540  5546 

5552  5558  5564  5570  5576  5582  5588  5594  5600  5606 

5612  5618  5625  5631  5637  5643  5649  5655  5661  5667 

5673  5679  5685  5691  5697  5703  5709  5715  5721  5727 


720  85733  85739  85745  85751  85757  85763  85769  85775  85781  85788 

5794  5800  5806  5812  5818  5824    5830  5836  5842  5848 

5854  5860  5866  5872  5878  5884  5890  5896  5902  5908 

5914  5920  5926  5932  5938  5944  5950  5956  5962  5968 

4  5974  5980  5986  5992  5998  6004  6010  6016  6022  6028 

5  6034  6040  6046  6052  6058  0064  6070  6076  6082  6088 

6  6094  6100  6106  6112  6118  6124  6130  6136  6141  6147 

7  6153  6159  6165  6171  6177  6183  6189  6195  6201  6207 

8  6213  6219  6225  6231  6237  6243  6249  6255  6261  6267 

9  6273  6279  6285  6291  6297  6303  6308  6314  6320  6326 

730  86332  86338  86344  86350  86356  86362  86368  86374  86380  86386 

1  6392  6398  6404  6410  6415  6421  6427  6433  6439  6445 

2  6451  6457  6463  6469  6475  6481  6487  6493  6499  6504 

3  6510  6516  6522  6528  6534  6540  6546  6552  6558  6564 

4  6570  6576  6581  6587  6593  6599  6605  6611  6617  6623 

5  6629  6635  6641  6646  6652  6658  6664  6670  6676  6682 

6  6688  6694  6700  6705  6711  6717  6723  6729  6735  6741 

7  6747  6753  6759  6764  6770  6776  6782  6788  6794  6800 

8  6806  6812  6817  6823  6829  6835  6841  6847  6853  6859 

9  6864  6870  6876  6882  6888  6894  6900  6906  6911  6917 

740  86923  86929  86935  86941  86947  86953  86958  86964  86970  86976 

1  6982  6988  6994  6999  7005  7011  7017  7023  7029  7035 

2  7040  7046  7052  7058  7064  7070  7075  7081  7087  7093 

3  7099  7105  7111  7116  7122  7128  7134  7140  7146  7151 

4  7157  7163  7169  7175  7181  7186  7192  7198  7204  7210 

5  7216  7221  7227  7233  7239  7245  7251  7256  7262  7268 

6  7274  7280  7286  7291  7297  7303  7309  7315  7320  7326 

7  7332  7338  7344  7349  7355  7361  7367  7373  7379  7384 

8  7390  7396  7402  7408  7413  7419  7425  7431  7437  7442 

9  7448  7454  7460  7466  7471  7477  7483  7489  7495  7500 

750  87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 


TABLE  VI.-LO(JAIUTIIMS  OF  NUMBERS     'iSr] 

N 

O    1   2    3   4 

5   6   7    8   9 

750 

87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 

1 

7564  7570  7570  7581  7587 

7593  7599  7604  7610  7616 

2 

7622  7628  7633  7639  7645 

7651  7656  7662  7668  7(574 

3 

7679  7685  7691  7697  7703 

7708  7714  7720  7726  7731 

4 

7737  7743  7749  7754  7760 

7766  7772  7777  7783  7789 

5 

7795  7800  7806  7812  7818 

7823  7829  7835  7841  7846 

6 

7852  7858  7864  7869  7875 

7881  7887  7892  7898  7904 

7 

7910  7915  7921  7927  7933 

7938  7944  7950  7955  79(51 

8 

7967  7973  7978  7984  7990 

7996  8001  8007  8013  8018 

9 

8024  8030  8030  8041  8047 

8053  8058  8064  8070  8076 

760 

88081  88087  88093  88098  88104  88110  88116  88121  88127  88133 

1 

8138  8144  8150  8156  8161 

8167  8173  8178  8184  8190 

2 

8195  8201  8207  8213  8218 

8224  8230  8235  8241  8247 

3 

8252  8258  8264  8270  8275 

8281  8287  8292  8298  8304 

4 

8309  8315  8321  8326  8332 

8338  8343  8349  8355  83(50 

5 

8366  8372  8377  8383  8389 

8395  8400  8406  8412  8417 

6 

8423  8429  8434  8440  8446 

8451  8457  8463  8468  8474 

7 

8480  8485  8491  8497  8502 

8508  8513  8519  8525  8530 

8 

8536  8542  8547  8553  8559 

8564  8570  8576  8581  8587 

9 

8593  8598  8604  8610  8615 

8621  8627  8632  8638  8643 

770 

88649  88655  88660  88666  88672  88677  88683  88689  88694  88700 

1 

8705  8711  8717  8722  8728 

8734  8739  8745  8750  8756 

2 

8762  8767  8773  8779  8784 

8790  8795  8801  8807  8812 

3 

8818  8824  8829  8835  8840 

8846  8852  8857  8863  8868 

4 

8874  8880  8885  8891  8897 

8902  8908  8913  8919  8925 

5 

8930  893()  8941  8947  8953 

8958  8964  8969  8975  8981 

6 

8986  8992  8997  9003  9009 

9014  9020  9025  9031  9037 

7 

9042  9048  9053  9059  9064 

9070  9076  9081  9087  9092 

8 

^098  9104  9109  9115  9120 

9126  9131  9137  9143  9148 

9 

9154  9159  9165  9170  9176 

9182  9187  9193  9198  9204 

780 

89209  89215  89221  89220  89232 

89237  89243  89248  89254  89260 

1 

9265  9271  9276  9282  9287 

9293  9298  9304  9310  9315 

2 

9321  932<>  9332  9337  9343 

9348  9354  9360  9365  9371 

3 

9376  9382  9387  9393  9398 

9404  9409  9415  9421  9426 

4 

9432  9437  9443  9448  9454 

9459  9465  9470  9476  9481 

5 

9487  9492  9498  9504  9509 

9515  9520  9526  9531  9537 

6 

9542  9548  9553  9559  9564 

9570  9575  9581  9586  9592 

7 

9597  9603  9609  9614  9620 

9625  9631  9636  9642  9647 

8 

9653  9658  9664  9669  9675 

9680  9686  9691  9697  9702 

9 

9708  9713  9719  9724  9730 

9735  9741  9746  9752  9757 

790 

89763  89768  89774  89779  89785  89790  89796  89801  89807  89812 

1 

9818  9823  9829  9834  9840 

9845  9851  9856  9862  9867 

2 

9873  9878  9883  9889  9894 

9900  9905  9911  9916  9922 

3 

9927  9933  9938  9944  9949 

9955  9960  9966  9971  9977 

4 

9982  9988  9993  9998  90004  90009  90015  90020  90026  90031 

5 

90037  90042  90048  90053  0059 

0064  0069  0075  0080  008(5 

6 

0091  0097  0102  0108  0113 

0119  0124  0129  0135  0140 

7 

0146  0151  0157  0162  0168 

0173  0179  0184  0189  0195 

8 

0200  0206  0211  0217  0222 

0227  0233  0238  0244  0249 

9 

0255  0260  02(56  0271  0276 

0282  0287  0293  0298  0304 

800 

90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 

•4  TACLE  VI.— LOGARITHMS   OF    XUMBERB. 


X 

O*       1        2        3 

4        5 

6        7        8        9 

800 

90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 

1 

0363    0369    0374    0380 

0385    0390 

0396    0401    0407    0412 

2 

0417    0423    0428    0434 

0439    0445 

0450    0455    0461    0466 

3 

0472    0477    0482    0488 

0493    0499 

0504    0509    0515    0520 

4 

0526    0531    0530    0542 

0547    0553 

0558    0563    0569    0574 

5 

0580    0585    0590    0596 

0601    0607 

0612    0617    0623    0628 

6 

0634    0639    0644    0650 

0655    0660 

0666    0671    0677    0682 

7 

0687    0693    0698    0703 

0709    0714 

0720    0725    0730    0736 

8 

0741    0747    0752    0757 

0763    0768 

0773    0779    0784    0789 

9 

0795    0800    0806    0811 

0816    0822 

0827    0832    0838    0843 

810 

90849  90854  90859  90865  90870  90875 

90881  90886  90891  90897 

1 

0902    0907    0913    0918 

0924    0929 

0934    0940    0945    0950 

2 

0956    0961    0966    0972 

0977    0982 

0988    0993    0998    1004 

3 

1009    1014    1020    1025 

1030    1036 

1041     1046    1052    1057 

4 

1062    1068    1073    1078 

1084    1089 

1094    1100    1105    1110 

5 

1116    1121    1126    1132 

1137    1142 

1148    1153    1158    1164 

6 

1169    1174    1180    1185 

1190    1196 

1201    1206    1212    1217 

.     7 

1222    1228    1233    1238 

1243    1249 

1254    1259    1265    1270 

8 

1275    1281    1286    1291 

1297    1302 

1307    1312    1318    1323 

9 

1328    1334    1339    1344 

1350    1355 

1360    1365    1371    1376 

820 

91381  91387  91392  91397  91403  91408  91413  91418  91424  91429 

1 

1434    1440    1445    1450 

1455    1461 

1466    1471    1477    1482 

2 

1487    1492    1498    1503 

1508    1514 

1519    1524    1529    1535 

3 

1540    1545    1551    1556 

1561    1566 

1572    1577    1582    1587 

4 

1593    1598    1603    1609 

1614    1619 

1624    1630    1635    1640 

5 

1645    1651    1656    1661 

1666    1672 

1677    1682    1687    1693 

6 

1698    1703    1709'   1714 

1719    1724 

1730    1735    1740    1745 

7 

1751    1756    1761    1766 

1772    1777 

1782    1787    1793    1798 

8 

1803    1808    1814    1819 

1824   J829 

1834    1840    1845    1850 

9 

1855    1861    1866    1871 

1876    1882 

1887    1892    1897    1903 

830 

91908  91913  91918  91924  91929  91934  91939  91944  91950  91955 

1 

1960    1965    1971    1976 

1981    1986 

1991    1997    2002    2007 

2 

2012    2018    2023    2028 

2033    2038 

2044    2049    2054    2059 

3 

2065    2070    2075    2080 

2085    2091 

2096    2101    2106    2111 

4 

2117    2122    2127    2132 

2137    2143 

2148    2153    2158    2163 

5 

2169    2174    2179    2184 

2189    2195 

2200    2205    2210    2215 

6 

2221    2226    2231    2236 

2241    2247 

2252    2257    2262    2267 

7 

2273    2278    2283    2288 

2293    2298 

2304    2309    2314    2319 

8 

2324    2330    2335    2340 

2345    2350 

2355    2361    2366    2371 

9 

2376    2381    2387    2392 

2397    2402 

2407    2412    2418    2423 

840 

92428  92433  92438  92443 

92449  92454  92459  92464  92469  92474 

1 

2480    2485    2490    2495 

2500    2505 

2511    2516    2521    2526 

2 

2531    2536    2542    2547 

2552    2557 

2562    2567    2572    2578 

3 

2583    2588    2593    2598 

2603    2609 

2614    2619    2624    2629 

4 

2634    2639    2645    2650 

2655    2660 

2665    2670    2675    2681 

5 

2686    2691    2696    2701 

2706    2711 

2716    2722    2727    2732 

6 

2737    2742    2747    2752 

2758    2763 

2768    2773    2778    2783 

7 

2788    2793    2799    2804 

2809    2814 

2819    2824    2829    2834 

2840    2845    2850    2855 

2860    2865 

2870    2875    2881    2886 

if 

2891    2896    2901    2906 

2911    2916 

2921    2927    2932    2937 

850 

92942  92947  92952  92957 

92962  92967 

92973  92978  92983  92988 

TABLE  Vr.  LOGARITHMS  OF  NUMBERS.     S35 

N 

01234567    89 

850 

92942  92947  92952  92957  92962  92967  92973  92978  92983  92988 

1 

2993  2998  3003  3008  3013  3018  3024  3029  3034  3039 

2 

3044  3049  3054  3059  3064  3069  3075  3080  3085  3090 

3 

3095  3100  3105  3110  3115  3120  3125  3131  3136  3141 

4 

3146  3151  3156  3161  3166  3171  3176  3181  3186  3192 

5 

3197  3202  3207  3212  3217  3222  3227  3232  3237  3242 

6 

3247  3252  3258  3263  3268  3273  3278  3283  3288  3293 

7 

3298  3303  3308  3313  3318  3323  3328  3334  3339  3344 

8 

3349  3354  3359  3304  3369  3374  3379  3384  3389  3394 

9 

3399  3404  3409  3414  3420  3425  3430  3435  3440  3445 

860 

93450  93455  93460  93465  93470  93475  93480  93485  93490  93495 

1 

3500  3505  3510  3515  3520  3526  3531  3536  3541  3546 

2 

3551  3556  3561  3566  3571  3576  3581  3586  3591  3596 

3 

3601  3606  3611  3616  3621  3626  3631  3636  3641  3646 

4 

3651  3656  3661  3666  3671  3676  3682  3687  3692  3697 

5 

3702  3707  3712  3717  3722  3727  3732  3737  3742  3747 

6 

3752  3757  3762  3767  3772  3777  3782  3787  3792  3797 

7 

3802  3807  3812  3817  3822  3827  3832  3837  3842  3847 

8 

3852  3857  3862  3867  3872  3877  3882  3887  3892  3897 

9 

3902  3907  3912  3917  3922  3927  3932  3937  3942  3947 

870 

93952  93957  93962  93967  93972  93977  93982  93987  93992  93997 

1 

4002  4007  4012  4017  4022  4027  4032  4037  4042  4047 

2 

4052  4057  4062  4067  4072  4077  4082  4086  4091  4096 

3 

4101  4106  4111  4116  4121  4126  4131  4136  4141  4146 

4 

4151  4156  4161  4166  4171  4176  4181  4186  4191  4196 

5 

4201  4206  4211  4216  4221  4226  4231  4236  4240  4245 

6 

4250  4255  4260  4265  4270  4275  4280  4285  4290  4295 

7 

4300  4305  4310  4315  4320  4325  4330  4335  4340  4345 

8 

4349  4354  4359  4364  4369  4374  4379  4384  4389  4394 

9 

4399  4404  4409  4414  4419  4424  4429  4433  4438  4443 

880 

94448  94453  94458  94463  94468  94473  94478  94483  94488  94493 

1 

4498  4503  4507  4512  4517  4522  4527  4532  4537  4542 

2 

4547  4552  4557  4562  4567  4571  4576  4581  4586  4591 

3 

4596  4601  4606  4611  4616  4621  4626  4630  4635  4640 

4 

4645  4650  4655  4660  4665  4670  4675  4680  4685  4689 

5 

4694  4699  4704  4709  4714  4719  4724  4729  4734  4738 

6 

4743  4748  4753  4758  4763  4768  4773  4778  4783  4787 

7 

4792  4797  4802  4807  4812  4817  4822  4827  4832  4836 

8 

4841  4846  4851  4856  4861  4866  4871  4876  4880  4885 

9 

4890  4895  4900  4905  4910  4915  4919  4924  4929  4934 

890 

94939  94944  94949  94954  94959  94963  94968  94973  94978  94983 

1 

4988  4993  4998  5002  5007  5012  5017  5022  5027  5032 

2 

5036  5041  5046  5051  5056  5061  5066  5071  5075  5080 

3 

5085  5090  5095  6100  5105  5109  5114  5119  5124  5129 

4 

5134  5139  5143  5148  5153  5158  5163  5168  5173  5177 

5 

5182  5187  5192  5197  5202  5207  5211  6216  6221  5226 

6 

5231  5236  5240  5245  5250  5255  6260  5265  5270  5274 

7 

5279  5284  6289  6294  5299  5303  5308  5313  6318  6323 

8 

6328  5332  6337  6342  5347  6362  5357  5361  6366  5371 

9 

5376  5381  6386  5390  6395  5400  6405  6410  6415  5419 

900 

95424  96429  96434  95439  95444  95448  96453  95458  96463  95468 

— 

236  TABLE  VI.— LOGARITHMS   OF  NUMBERS. 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900 

95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 

1 

5472 

5477 

5482 

5487 

5492 

5497 

5501 

5506 

5511 

5516 

2 

5521 

5525 

5530 

5535 

5540 

5545 

5550 

5554 

5559 

6664 

3 

5569 

5574 

5578 

5583 

5588 

5593 

5598 

5602 

5607 

6612 

4 

5617 

5622 

5626 

5631 

5636 

5641 

5646 

5650 

5655 

6660 

5 

5665 

6670 

5674 

5679 

5684 

5689 

5694 

5698 

6703 

6708 

6 

5713 

5718 

5722 

5727 

5732 

5737 

5742 

5746 

5751 

6756 

7 

5761 

5766 

5770 

5775 

5780 

5785 

5789 

5794 

5799 

5804 

8 

5809 

5813 

5818 

5823 

5828 

5832 

5837 

5842 

5847 

6852 

9 

5856 

5861 

5866 

5871 

5875 

5880 

5885 

5890 

5895 

6899 

910 

95904  95909  95914  95918  95923  95928  95933  95938  95942  95947 

1 

5952 

5957 

5961 

5966 

5971 

5976 

5980 

5985 

5990 

5995 

2 

5999 

6004 

6009 

6014 

6019 

6023 

6028 

6033 

6038 

6042 

3 

6047 

6052 

6057 

6061 

6066 

6071 

6076 

6080 

6085 

6090 

4 

6095 

6099 

6104 

6109 

6114 

6118 

6123 

6128 

6133 

6137 

5 

6142 

6147 

6152 

6156 

6161 

6166 

6171 

6175 

6180 

6185 

6 

6190 

6194 

6199 

6204 

6209 

6213 

6218 

6223 

6227 

6232 

7 

6237 

6242 

6246 

6251 

6256 

6261 

6265 

6270 

6275 

6280 

8 

6284 

6289 

6294 

6298 

6303 

6308 

6313 

6317 

6322 

6327 

9 

6332 

6336 

6341 

6346 

6350 

6355 

6360 

6365 

6369 

6374 

920 

96379  96384  96388  96393 

96398  96402  96407  96412 

96417  96421 

1 

6426 

6431 

6435 

6440 

6445 

6450 

6454 

6459 

6464 

6468 

2 

6473 

6478 

6483 

6487 

6492 

6497 

6501 

6506 

6511 

6515 

3 

6520 

6525 

6530 

6534 

6539 

6544 

6548 

6553 

6558 

6562 

4 

6567 

6572 

6577 

6581 

6586 

6591 

6595 

6600 

6605 

6609 

5 

6614 

6619 

6624 

6628 

6633 

6638 

6642 

6647 

6652 

6656 

6 

6661 

6666 

6670 

6675 

6680 

6685 

6689 

6694 

6699 

6703 

7 

6708 

6713 

6717 

6722 

6727 

6731 

6736 

6741 

6745 

6750 

8 

6755 

6759 

6764 

6769 

6774 

6778 

6783 

6788 

6792 

6797 

9 

6802 

6806 

6811 

6816 

6820 

6825 

6830 

6834 

6839 

6844 

930 

96848  96853  96858  96862  96867  96872  96876  96881  96886  96890 

1 

6895 

6900 

6904 

6909 

6914 

6918 

6923 

6928 

6932 

6937 

2 

6942 

6946 

6951 

6956 

6960 

6965 

6970 

6974 

6979 

6984 

3 

■6988 

6993 

6997 

7002 

7007 

7011 

7016 

7021 

7025 

7030 

4 

7035 

7039 

7044 

7049 

7053 

7058 

7063 

7067 

7072 

7077 

5 

7081 

7086 

7090 

7095 

7100 

7104 

7109 

7114 

7118 

7123 

6 

7128 

7132 

7137 

7142 

7146 

7151 

7155 

7160 

7165 

7169 

7 

7174 

7179 

7183 

7188 

7192 

7197 

7202 

7206 

7211 

7216 

8 

7220 

7225 

7230 

7234 

7239 

7243 

7248 

7253 

7257 

7262 

9 

7267 

7271 

7276 

7280 

7285 

7290 

7294 

7299 

7304 

7308 

940 

97313  97317  97322  97327  97331  97336  97340  97345  97350  97354 

1 

7359 

7364 

7368 

7373 

7377 

7382 

7387 

7391 

7396 

7400 

2 

7405 

7410 

7414 

7419 

7424 

7428 

7433 

7437 

7442 

7447 

3 

7451 

7456 

7460 

7465 

7470 

7474 

7479 

7483 

7488 

7493 

4 

7497 

7502 

7506 

7511 

7516 

7520 

7525 

7529 

7534 

7539 

5 

7543 

7548 

7552 

7557 

7562 

7566 

7571 

7575 

7580 

7585 

6 

7589 

7594 

7598 

7603 

7607 

7612 

7617 

7621 

7626 

7630 

7 

7635 

7640 

7644 

7649 

7653 

7658 

7663 

7667 

7672 

7676 

8 

7681 

7685 

7690 

7695 

7699 

7704 

7708 

7713 

7717 

7722 

9 

7727 

7731 

7736 

7740 

7745 

7749 

7754 

7759 

7763 

7768 

950 

97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 

TABLE  VT.  LdGA'RITILAfB  OF  NTAniKUS.'    337 

N 

0123456789 

050 

97772  97777  97782  97786  97791  9779;-)  97800  97804  97809  97813 

1 

7818  7823  7827  7832  7836  7841  7845  7850  7855  7859 

2 

7864  7808  7873  7877  7882  7886  7891  7896  7900  7905 

3 

7909  7914  7918  7923  7928  7932  7937  7941  7946  7950 

4 

7955  7959  7964  7968  7973  7978  7982  7987  7991  7996 

5 

8000  8005  8009  8014  8019  8023  8028  8032  8037  8041 

6 

8046  8050  8055  8059  8064  8068  8073  8078  8082  8087 

7 

8091  8096  8100  8105  8109  8114  8118  8123  8127  8132 

8 

8137  8141  8146  8150  8155  8159  8164  8168  8173  8177 

9 

8182  8186  8191  8195  8200  8204  8209  8214  8218  8223 

960 

98227  98232  98236  98241  98245  98250  98254  98259  98263  98268 

1 

8272  8277  8281  8286  8290  8295  8299  8304  8308  8313 

2 

8318  8322  8327  8331  8336  8340  8345  8349  8354  8358 

3 

8363  8367  8372  8376  8381  8385  8390  8394  8399  8403 

4 

8408  8412  8417  8421  8426  8430  8435  8439  8444  8448 

5 

8453  8457  8462  8466  8471  8475  8480  8484  8489  8493 

6 

8498  8502  8507  8511  8516  8520  8525  8529  8534  8538 

7 

8543  8547  8552  8556  8561  8565  8570  8574  8579  8583 

8 

8588  8592  8597  8601  8605  8610  8614  8619  8623  8628 

9 

8632  8637  8641  8646  8650  8655  8659  8664  8668  8673 

D70 

98677  98682  98686  98691  98695  98700  98704  98709  98713  98717 

1 

,  8722  8726  8731  8735  8740  8744  8749  8753  8758  8762 

2 

8767  8771  8776  8780  8784  8789  8793  8798  8802  8807 

3 

8811  8816  8820  8825  8829  8834  8838  8843  8847  8851 

4 

8856  8860  8865  8869  8874  8878 '  8883  8887  8892  8896 

5 

8900  8905  8909  8914  8918  8923  8927  8932  8936  8941 

6 

8945  8949  8954  8958  8963  8967  8972  8976  8981  8985 

7 

8989  8994  8998  9003  9007  9012  9016  9021  9025  9029 

8 

9034  9038  9043  9047  9052  9056  9061  9065  9069  9074 

9 

9078  9083  9087  9092  9096  9100  9105  9109  9114  9118 

980 

99123  99127  99131  99136  99140  99145  99149  99154  99158  99162 

1 

9167  9171  9176  9180  9185  9189  9193  9198  9202  9207 

2 

9211  9216  9220  9224  9229  9233  9238  9242  9247  9251 

3 

9255  9260  9264  9269  9273  9277  9282  9286  9291  9295 

4 

9300  9304  9308  9313  9317  9322  9326  9330  9335  9339 

5 

9344  9348  9352  9357  9361  9366  9370  9374  9379  9383 

6 

9388  9392  9396  9401  9405  9410  9414  9419  9423  9427 

7 

9432  9436  9441  9445  9449  9454  9458  9463  9467  9471 

8 

9476  9480  9484  9489  9493  9498  9502  9506  9511  9515 

9 

9520  9524  9528  9533  9537  9542  9546  9550  9555  9559 

990 

99564  99568  99572  99577  99581  99585  99590  99594  99599  99603 

1 

9607  9612  9616  9621  9625  9629  9634  9638  9642  9647 

2 

9()51  9656  9660  9664  9669  9673  9677  9682  9686  9691 

3 

9695  9699  9704  9708  9712  9717  9721  9726  9730  9734 

4 

9739  9743  9747  9752  9756  9760  9765  9769  9774  9778 

5 

9782  9787  9791  9795  9800  9804  9808  9813  9817  9822 

6 

9826  9830  9835  9839  9843  9848  9852  9856  9861  98(55 

7 

9870  9874  9878  9883  9887  9891  9896  9900  9904  9909 

8 

9913  9917  9922  9926  9930  9935  9939  9944  9948  9952 

9 

9957  9961  9965  9970  9974  9978  9983  9987  9991  9996 

1000 

1 

00000  00004  00009  00013  00017  00022  00026  00030  00035  00039 

t 

3 

1-  •  t 


Vv^nm.Cm.  Sin,  Cos.  iit^ 

ni  92 

238  TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES. 


/ 

Sine 

0° 

1« 

2° 

1 
/ 

Cosine 

Sine  - 

Cosine 

Sine 

Cosine 

0 

— ao 

10.00000 

8.24186 

9.99993 

8.54^82 
54642 

9.99974 

60   . 

1 

6.46373 

00000 

24903 

99993 

99973 

59 

2 

76476 

00000 

25609 

99993 

54999  , 

99973 

58 

3 

94085 

00000 

26304 

99993 

55354 

99972 

57 

4 

7.06579 

00000 

26988 

99992 

55705 

99972 

56 

5 

16270 

00000 

27661 

99992 

56054 

69971 

55 

6 

24188 

00000 

28324 

99992 

56400 

99971 

54 

7 

30882 

00000 

28977 

99992  • 

56743 

99970 

53 

8 

36682 

00000 

29621 

99992 

57084 

99970 

52 

9 

41797 

00000 

30255 

9999] 

57421 

99969 

51 

10 

7.46373 

10.00000 

8.30879 

9.99991 

8.57757 

9.99969 

50 

11 

50512 

00000 

31495 

99991 

58089 

99968 

49 

12 

54291 

00000 

32103 

99990 

58419 

99968 

48 

13 

57767 

ooooo 

32702 

99990 

58747 

99967 

47 

14 

60985 

00000 

33292 

99990 

59072 

99967 

46 

15 

63982 

ooooo 

33875 

99990 

59395 

99967 

45 

16 

66784 

■  ooooo 

34450 

99989 

59715 

99960 

44 

17 

69417 

9.99999 

35018 

99989 

60033 

99966 

43 

18 

71900 

99999 

35578 

99989 

60349 

99965 

42 

19 

74248 

99999 

36131 

99989 

60662 

99964 

41 

20 

7.76475 

9.99999 

8.36678 

9.99988 

8.60973 

9.99964 

40 

21 

78594 

99999 

37217 

99988  • 

61282 

99963 

39 

22 

80615 

99999 

37750 

99988 

61589 

99963 

38 

23 

82545 

99999 

38276 

99987 

61894 

99962 

37 

24 

84393 

99999 

.38796 

99987 

62196 

99962 

36 

25 

86166 

99999 

39310 

99987 

62497 

99961 

35 

26 

87870 

99999 

39818 

99986 

62795 

99961 

34 

27 

89509 

99999 

40320 

99986 

63091 

99960 

33 

28 

91088 

99999 

40816 

99986 

63385 

99960 

32 

29 

92612 

99998 

41307 

99985 

63678 

99959 

31 

30 

7.94084 

9.99998 

8.41792 

9.99985 

8.63968 

9.99959 

30 

31 

95508 

99998 

42272 

9998E 

64256 

99958 

29 

32 

96887 

99998 

42746 

99984 

64543 

99958 

28 

33 

98223 

99998 

43216 

99984 

64827 

99957 

27 

34 

99520 

99998 

43680 

99984 

65110 

99956 

26 

35 

8.00779 

99998 

44139 

99983 

65391 

99956 

25 

36 

02002 

99998 

44594 

99983 

65670 

99955 

24 

37 

03192 

99997 

45044 

99983 

65947 

99955 

23 

38 

04350 

99997 

45489 

99982 

66223 

99954 

22 

39 

05478 

99997 

45930 

99982 

66497 

99954 

21 

40 

8.06578 

9.99997 

8.46366 

9.99982 

8.66769 

9.99953 

20 

41 

07650 

99997 

46799 

99981 

67039 

99952 

19 

42 

08696 

99997 

47226 

99981 

67308 

99952 

1* 

43 

09718 

99997 

47650 

99981 

67575 

99951 

17 

44 

10717 

99996 

48069 

99980 

67841 

99951 

16 

45 

11693 

99996 

48485 

99980 

68104 

99950 

15 

46 

12647 

99996 

48896 

99979 

68367 

99949 

14 

47 

13581 

99996 

49304 

99979 

68627 

99949 

13 

48 

14495 

99996 

49708 

99979 

68886 

99948 

12 

49 

15391 

99996 

50108 

99978 

69144 

99948 

11 

50 

8.16268 

9.99995 

8.50504 

9.99978 

8.69400 

9.99947 

10 

51 

17128 

99995 

50897 

99977 

69654 

99946 

9 

52 

17971 

99995 

51287 

99977 

69907 

99946 

8 

53 

18798 

99995 

51673 

99977 

70159 

99945 

7 

54 

19610 

99995 

52055 

99976 

70409 

99944 

6 

55 

20407 

99994 

52434 

99976 

70658 

99944 

5 

56 

21189 

99994 

52810 

99975 

70905 

99943 

4 

57 

21958 

99994 

53183 

99975 

71151 

99942 

3 

58 

22713 

99994 

53552 

99974 

71395 

99942 

2 

59 

23456 

99994 

53919 

99974 

71638 

99941 

1 

60 

24186 

99993 

54282 

99974 

71880 

99940 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

89° 

8.« 

o 

87 
»— 

0 

.^/-r^ 

S.^-iv 

1  - 

T — 

.  Sin 

:  Cn& 

93  94  95 

TABLE  VII.— LOGARITHMIC   SINES  AND  COSINES.    239 


/ 

3° 

4° 

6° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

8.71880 

9.99940 

8.843.58 

9.99894 

8.94030 

9.99834 

60 

1 

72120 

99940 

84539 

99893 

94174 

99833 

59 

2 

72359 

99939 

84718 

99892 

94317 

99832 

58 

3 

72597 

99938 

84897 . 

99891 

94461 

99831 

.57 

4 

72834 

9993S 

85075 

99891 

94603 

99830 

56 

5 

73069 

99937 

85252 

99890 

94746 

99829 

55 

6 

73303 

99936 

85429 

59889 

94887 

99828 

54 

7 

73535 

99936 

85605 

99888 

95029 

99827 

53 

8 

73767 

99935 

85780 

99887 

95170 

99825 

52 

9 

73997 

99934 

85955 

99886 

95310 

99824 

51 

10 

8.74226 

9.99934 

8.86128 

9.99885 

8.954.50 

9.99823- 

50 

11 

74454 

99933 

86301 

99884 

95589 

99822 

49 

12 

74680 

99932 

86474 

99883 

95728 

99821 

48 

13 

74906 

99932 

86645 

99882 

95867 

99820 

47 

14 

75130 

99931 

86816 

99881 

96005 

99819 

46 

15 

75353 

99930 

86987 

99880 

96143 

99817 

45 

16 

75575 

99929 

87156 

99879 

96280 

99816 

44 

17 

75795 

99929 

87325 

99879 

96417 

99815 

43 

18 

76015 

99928 

87494 

99878 

96553 

99814 

42 

19 

76234 

99927 

87661 

99877 

96689 

99813 

41 

20 

8.76451 

9.99926 

8.87829 

9.99876 

8.96825 

9.99812 

40 

21 

76667 

99926 

87995 

99875 

96960 

99810 

39 

22 

76883 

99925 

88161 

9987'4 

97095 

99809 

38 

23 

77097 

99924 

88326 

99873 

97229 

99808 

37 

24 

77310 

99923 

88490 

99872 

97363 

99807 

36 

25 

77522 

99923 

88654 

99871 

97496 

99806 

35 

26 

77733 

99922 

88817 

99870 

97629 

99804 

34 

27 

77943 

999^1 

88980 

99869 

97762 

99803 

33 

28 

78152 

99920 

89142 

99868 

97894 

99802 

32 

29 

78360 

99920 

89304 

99867 

98026 

90801 

31 

30 

8.78568 

9.99919 

8.89464 

9.99866 

8.981.57 

9.99800 

30 

31 

78774 

99918 

89625 

99865 

98288 

99798 

29 

32 

78979 

99917 

89784 

99864 

98419 

99797 

28 

33 

79183 

99917 

89943 

99863 

98549 

99796 

27 

34 

79386 

99916 

90102 

99862 

98679 

99795 

26 

35 

79588 

99915 

90260 

99861 

98808 

99793 

25 

36 

79789 

99914 

90417 

99860 

98937 

99792 

24 

37 

79990 

99913 

90574 

99859 

99066 

99791 

23 

38 

80189 

99913 

90730 

99858 

99194 

99790 

22 

39 

80388 

99912 

90885 

99857 

99322 

99788 

21 

40 

8.80585 

9.99911 

8.91040 

9.99856 

8.99450 

-9.99787 

20 

41 

80782 

99910 

91195 

99855 

99577 

99786 

19 

42 

80978 

99909 

91349 

99854 

99704 

99785 

18 

43 

81173 

99909 

91502 

99853 

99830 

99783 

17 

44 

81367 

99908 

91655 

99852 

99956 

99782 

16 

45 

81560 

99907 

91807 

99851 

9.00082 

99781 

15 

46 

81752 

99906 

91959 

99850 

00207 

99780 

14 

47 

81944 

99905 

92110 

99848 

00332 

99778 

13 

48 

82134 

99904 

92261 

99847 

00456 

99777 

12 

49 

82324 

99904 

92411 

99846 

00581 

99776 

11 

50 

8.82513 

9.99903 

8.92561 

9.99845 

9.00704 

9.99775 

10 

51 

82701 

99902 

92710 

99844 

00828 

99773 

9 

52 

82888 

99901 

92a59 

99843 

00951 

99772 

8 

53 

83075 

99900 

93007 

99842 

01074 

99771 

7 

54 

83261 

99899 

93154 

99841 

01196 

99769. 

6 

55 

83446 

99898 

93301 

99840 

01318 

99768 

5 

56 

83630 

99898 

93448 

99839 

01440 

99767 

4 

57 

83813 

99897 

93594 

99838 

01.561 

997()5 

3 

58 

83996 

99896 

93740 

99a37 

01682 

99764 

o 

59 

84177 

99895 

9.3885 

99836 

01803 

99763 

1 

60 

84358 

99894 

94030 

99834 

01923 

99761 

0 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

^ 4 . 

86° 

86° 

84° 

^^nL'^)i^. -j^r^WA,.  ^'^rJR!. 


•  r 


96 


•>- 


jr*-: 


yb  9  7  QQ 

340   TABLE  VII.— LOGARITHMIC  SINES  AND  COSINES. 


/ 

6° 

7° 

8» 

/ 

Sine 
9.01923 

Cosine 

Sine 

Cosine 

Sine 
9.14356 

Cosine 
9.99575 

0 

9.99761 

9.08589 

9.99675 

fiO 

1 

02043 

99760 

08692 

99674 

14445 

99574 

Kf\J 

59 

58 

o 

02163 

99759 

08795 

99672 

14535 

99572 

3 

02283 

99757 

08897 

99670 

14624 

99570 

57 
56 

4 

02402 

99756 

08999 

99669 

14714 

99568 

5 

02520 

99755 

09101 

99667 

14803 

99566 

55 
54 

-  6 

02639 

99753 

09202 

99666 

14891 

99565 

7 

02757 

99752 

09304 

99664 

14980 

99563 

53 

8 

02874 

99751 

09405 

99663 

15069 

99561 

5" 

9 

02992 

99749 

09506 

99661 

15157 

99559 

51 

10 

9.03109 

9.99748 

9.09606 

9.99659 

9.15245 

9.99557 

50 

11 

03226 

99747 

09707 

99658 

15333 

99556 

49 

12 

03342 

99745 

09807 

99656 

15421 

99554 

48 

13 

03458 

99744 

09907 

99655 

15508 

99552 

47 

14 

03574 

99742 

10006 

99653 

15596 

99550 

46 

15 

03690 

99741 

10106 

99651 

15683 

99548 

45 

16 

03805 

99740 

10205 

99650 

15770 

99546 

44 

17 

03920 

99738 

10304 

99648 

15857 

995i5 

43 

18 

040:34 

99737 

10402 

99647 

15944 

99543 

42 

19 

04149 

99736 

10501 

99645 

16030 

99541 

41 

20 

9.04262 

9.99734 

9.10599 

9.99643 

9.16116 

9.995:39 

40 

21 

04376 

99733 

10697 

99642 

16203 

995:37 

39 

22 

04490 

99731 

10795 

99640 

16289 

99535 

38 

23 

04603 

99730 

10893 

99638 

16374 

99533 

37 

24 

04715 

99728 

10990 

99637 

16460 

99532 

36 

2o 

04828 

99727 

11087 

99635 

16545 

99530 

35 

26 

04940 

99726 

11184 

99633 

16631 

99528 

34 

27 

05052 

99724 

11281 

99632 

16716 

99526 

33 

28 

05164 

99723 

11377 

99630 

16801 

99524 

32 

29 

05275 

99721 

11474 

99629 

16886 

99522 

31 

30 

9.05386 

9.99720 

9.11570 

9.99627 

9.16970 

9.99520 

30 

31 

05497 

99718 

11C66 

99625 

17055' 

99518 

29 

32 

05607 

99717 

11761 

99624 

17139 

99517 

28 

33 

05717 

99716 

11857 

99622 

17223 

995J5 

27 

34 

05827 

99714 

11952 

99620 

17307 

99513 

26 

:ir, 

05937 

90713 

12047 

9961 S 

17:391 

99511 

25 

36 

06046 

99711 

12142 

99617 

17474 

99509 

24 

37 

06155 

99710 

12236 

99615 

17558 

99507 

23 

38 

06264 

99708 

12331 

99613 

17641 

99505 

22 

39 

06372 

99707 

12425 

99612 

17724 

99503 

21 

40 

9.06481 

9.99705 

9.12519 

9.99610 

9.17807 

9.99501 

20 

41 

06589 

99704 

12612 

99608 

17890 

99499 

19 

42 

06696 

99702 

12706 

99607 

17973 

99497 

18 

43 

06804 

99701 

12799 

99605 

1805.-) 

99495 

17 

44 

06911" 

99699 

12892 

99603 

18137 

99494 

16 

45 

07018 

99698 

12985 

99601 

18220 

99492 

15 

46 

07124 

99696 

13078 

99600 

18:302 

99490 

14 

47 

072;:J1 

99695 

13171 

99598 

ia383 

99488 

13 

48 

07337 

99693 

13263 

99596 

18465 

994^6 

12 

49 

07442 

99692 

13:355 

99595 

18547 

99484 

11 

50 

9.07548 

9.99690 

9.13447 

9.99.593 

9.18628 

9.99482 

10 

51 

07653 

99689 

13539 

99591 

18709 

99480 

9 

52 

07758 

99687 

136:30 

995S9 

18790 

99478 

8 

53 

07863 

99686 

13722 

99588 

18871 

99476 

i 

54 

07968 

99684 

13813 

995S6 

18952 

99474 

6 

55 

08072 

99683 

13904 

99584 

190:33 

99472 

5 

56 

08176 

99681 

13994 

99582 

19113 

99470 

4 

57 

08280 

99680 

14085 

99581 

19193 

99468 

3 

58 

08383 

99678 

14175 

99579 

19273 

99466 

2 

59 

08486 

99677 

14266 

99577 

19:353 

99464 

1 

60 

08589 

99675 

14:356 

99575 

19433 

99462 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sinp 

/ 

83° 

82° 

81"      ' 

Si^tJ^a.  ^n^^^M  siti, 


r:. 


£.% 


TABLE  VII.— LOGARITIIMIC   SINES  AND  COSINES.  241 


/ 

9 

o 

10 

o 

11 

o 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.19433 

9.99462 

9.23967 

9.993.35 

9.28060 

9.99195 

60 

1 

19513 

99460 

24039 

99333 

28125 

99192 

59 

2 

19592 

99458 

24110 

99331 

28190 

99190 

58 

3 

19672 

99456 

24181 

99328 

28254 

99187 

57 

4 

19751 

99454 

24253 

99326 

28319 

99185 

56 

5 

19830 

99452 

24324 

99324 

28384 

99182 

55 

6 

19909 

99450 

24395 

99322 

28448 

99180 

54 

i 

19988 

99448 

24466 

99319 

28512 

99177 

53 

8 

20067 

99446 

24536 

99317 

28577 

99175 

52 

9 

20145 

99444 

24607 

99315 

28641 

99172 

51 

10 

9.20223 

9.99442 

9.24677 

9.99.313 

9.28705 

9.99170 

50 

11 

20302 

99440 

24748 

99310 

28769 

99167 

49 

12 

20380 

99438 

24818 

99308 

28833 

99165 

48 

13 

20458 

99436 

24888 

99306 

28896 

99162 

47 

14 

20535 

99434 

24958 

99.304 

28960 

99160 

46 

15 

20613 

99432 

25028 

99301 

29024 

991.57 

45 

16 

20691 

99429 

25098 

99299 

29087 

99155 

44 

17 

207G8 

99427 

25168 

99297 

29150 

99152 

43 

18 

20845 

99425 

25237 

99294 

29214 

991.50 

42 

19 

20922 

99423 

25307 

99292 

29277 

99147 

41 

20 

9.20999 

9.99421 

9.25376 

9.99290 

9.29340 

9.99145 

40 

21 

21076 

99419 

25445 

99288 

29403 

99142 

39 

22 

21153 

99417 

25514 

99285 

29466 

99140 

38 

23 

21229 

99415 

25583 

99283 

29529 

991.37 

37 

24 

21306 

99413 

25652 

99281 

29591 

99135 

36 

25 

21382 

99411 

25721 

99278 

29654 

99132 

35 

26 

21458 

99409 

25790 

99276 

29716 

99130 

34 

27 

21534 

99407 

25858 

99274 

29779 

99127 

33 

28 

21610 

99404 

25927 

99271 

29841 

99124 

32 

29 

21685 

99402 

25995 

99269 

29903 

99122 

31 

30 

9.21761 

9.99400 

9.26063 

9.99267 

9.29966 

9.99119 

30 

31 

21836 

99398 

26131 

99264 

30028 

99117 

29 

32 

21912 

99396 

26199 

99262 

30090 

99114 

28 

33 

21987 

99394 

26267 

99260 

30151 

99112 

27 

34 

22062 

99892 

26335 

99257 

30213 

99109 

26 

35 

22137 

99390 

26403 

99255 

30275 

99106 

25 

36 

22211 

99.388 

26470 

99252 

30336 

99104 

24 

37 

22^86 

99385 

26538 

99250 

30398 

99001 

23 

38 

22361 

99383 

26605 

99248 

30459 

99099 

22 

39 

22435 

99381 

26672 

99245 

30521 

99096 

21 

40 

9.22509 

9.99379 

9.267.39 

9.99243 

9.30582 

9.99093 

20 

41 

22583 

99377 

26806 

99241 

30643 

99091 

19 

42 

22657 

99375 

26873 

99238 

30704 

99088 

18 

43 

22731 

99372 

26940 

99236 

30765 

99086 

17 

44 

22805 

99370 

27007 

99233 

30826 

99083 

16 

45 

22878 

99368 

27073 

99231 

30887 

99080 

15 

46 

22952 

99.366 

27140 

99229 

30947 

99078 

14 

47 

23025 

99364 

27206 

99226 

31008 

99075 

13 

48 

23098 

99362 

27273 

99224 

31068 

99072 

12 

49 

23171 

99359 

27339 

99221 

31129 

99070 

11 

50 

9.23244 

9.99357 

9.27405 

9.99219 

9.31189 

9.99067 

10 

51 

23317 

99355 

27471 

99217 

31250 

99064 

9 

52 

23390 

99353 

27537 

99214 

31310 

99062 

8 

53 

23462 

99351 

27602 

99212 

31370 

99059 

7 

54 

23535 

99348 

27668 

99209 

31430 

99056 

6 

55 

23607 

99346 

27734 

99207 

31490 

99054 

5 

56 

23679 

99344 

27799 

99204 

31549 

99051 

4 

57 

23752 

99342 

27864 

99202 

31609 

99048 

3 

58 

23823 

99340 

27930 

99200 

31669 

99046 

2 

59 

23895 

99337 

27995 

99197 

.31728 

99043 

1 

60 

23967 

99335 

28060 

99195 

31788 

99040 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

80O 

79° 

78° 

1 

i 

70 

Jl 

G9 

JL 

GS 

^0^^. 

1.  ,   V—   V.^  t-- 

4 

v.n  t 


.nr 


mmiC   SINES  AND 


242    TABLE   VII.— LOG AK 


COSINES. 


/ 

12° 

13° 

14° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.31788 

9.99040 

9.35209 

9.98872 

9.38368 

9.98690 

60 

1 

31847 

99038 

35263 

98869 

38418 

98687 

59 

2 

31907 

99035 

35318 

98867 

38469 

98684 

58 

3 

31966 

99032 

35373 

98864 

38519 

98681 

57 

4 

32025 

99030 

35427 

98861 

38570 

98678 

56 

5 

32084 

99027 

35481 

98858 

38620 

98675 

55 

6 

32143 

99024 

35536 

98855 

38670 

98671 

54 

7 

32202 

99022 

35590 

98852 

38721 

98668 

53 

8 

32261 

99019 

35644 

98849 

38771 

98665 

52 

9 

32319 

99016 

35698 

98846 

38821 

98652 

51 

10 

9.32378 

9.99013 

9.35752 

9.98843 

9.38871 

9.98659 

50 

11 

32437 

99011 

35806 

98840 

38921 

98656 

49 

12 

32495 

99008 

35860 

98837 

38971 

98652 

48 

13 

32553 

99005 

35914 

98834 

39021 

98649 

47 

14 

32612 

99002 

35968 

98»^1 

39071 

98646 

46 

15 

32670 

99000 

36022 

98828 

39121 

98643 

45 

16 

32728 

98997 

36075 

98825 

39170 

98640 

44 

17 

32786 

98994 

36129 

98822 

39220 

98636 

43 

18 

32844 

98991 

36182 

98819 

39270 

98633 

42 

19 

32902 

98989 

36336 

98816 

39319 

98';30 

41 

20 

9.32960 

9.98986 

9.36289 

9.98813 

9.39369 

9.98627 

40 

21 

33018 

98983 

36342 

98810 

39418 

98623 

39 

22 

33075 

9S980 

36395 

98807 

39467 

98620 

38 

28 

33133 

98978 

36449 

98804 

39517 

98617 

37 

24 

33190 

98975 

36502 

98801 

39566 

98614 

36 

25 

33248 

98972 

36555 

98798 

39815 

98610 

35 

26 

33305 

98969 

36608 

98795 

39664 

98607 

34 

27 

33362 

98967 

36660 

98792 

39713 

98604 

33 

28 

33420 

98964 

36713 

98789 

39762 

98601 

32 

29 

33477 

98961 

36766 

98786 

39811 

98597 

31 

30 

9.33534 

9.98958 

9.36819 

9.98783 

9.39860 

9.98594 

30 

31 

3359] 

98955 

36871 

98780 

39909 

98591 

29 

32 

33647 

98953 

36924 

98777 

39958 

98588 

88 

33 

33704 

98950 

36976 

98774 

40006 

98584 

27 

34 

33761 

98947 

37028 

98771 

40055 

98581 

26 

35 

33818 

98944 

37081 

98768 

40103 

98578 

25 

36 

33874 

98941 

37133 

98765 

40152 

98574 

24 

37 

38931 

98938 

37185 

98762 

40200 

98571 

23 

38 

33987 

98936 

37237 

98759 

40249 

98568 

22 

39 

34043 

98933 

37289 

98756 

40297 

98565 

21 

40 

9.34100 

9.98930 

9.37341 

9.98753 

9.40346 

9.98561 

20 

41 

34156 

9S927 

37393 

98750 

40394 

98558 

19 

42 

34212 

98924  > 

37445 

98746 

40442 

98^55 

18 

43 

34268 

98921 

37497 

98743 

40490 

98551 

17 

44 

34324 

98919 

37549 

98740 

40538 

98548 

16 

45 

34380 

98916 

37600 

98737 

40586 

98545 

15 

46 

34436 

98913 

87652 

98734 

40634 

98541 

14 

47 

34491 

98910 

37703 

98731 

40682 

98538 

13 

48 

34547 

98907 

37755 

98728 

40730 

98535 

12 

49 

34602 

98904 

37806 

98725 

40778 

98531 

11 

50 

9.34658 

9.98901 

9.37858 

9.98722 

9.40825 

9.98528 

10 

51 

34713 

98898 

37909 

98719 

40873 

98525 

9 

52 

34769 

98896 

37960 

98715 

40921 

98521 

8 

53 

34824 

98893 

38011 

98712 

40968 

98518 

1 

54 

34879 

98890 

38062 

98709 

41016 

98515 

6 

55 

34934 

98887 

38113 

98706 

41063 

98.M1 

5 

56 

34989 

98884 

38164 

98703 

41111 

98508 

4 

57 

35044 

988*^1 

38215 

98700 

41158 

98505 

3 

58 

35099 

98878 

38266 

98697 

41205 

98501 

2 

59 

35154 

98875 

38317 

98694 

41252 

98498 

1 

60 

35209 

98872 

38368 

98690 

41300 

98494 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

77° 

76° 

75° 

A3^ 


6i: 


O**.      A^/i^ 


6 


PnQ 


i-f) ;)    . 


•:• 


TABLE  ^hr  — LOGARlTlhnC   SINES  AN^)  TOSINES.    243 


, 

1 

5° 

16 

0 

17° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.41300 

9.98494 

9.44034 

9.98284 

9.46594 

9.98060 

60 

1 

41347 

98491 

44078 

98281 

46635 

98050 

59 

2 

41394 

98488 

44122 

98277 

46676 

98052 

58 

3 

41441 

98484 

44166 

98273 

46717 

98048 

57 

4 

41488 

98481 

44210 

98270 

467.58 

98044 

56 

5 

41535 

98477 

44253 

98266 

46800 

98040 

55 

6 

41582 

98474 

44297 

98262 

46841 

98030 

54 

7 

41028 

98471 

44341 

98259 

46882 

98032 

53 

8 

41675 

98467 

44385 

98255 

46923 

98029 

52 

9 

41722 

98464 

44428 

98251 

46964 

98025 

51 

10 

9.41768 

9.98460 

9.44472 

9.98248 

9.47005 

9.98021 

50 

11 

41815 

98457 

44516 

98244 

47045 

98017 

49 

12 

41861 

98453 

44559 

98240 

47086 

98013 

48 

13 

41908 

98450 

44602 

98237 

47127 

98009 

47 

14 

41954 

98447 

44646 

98233 

47168 

98005 

46 

15 

42001 

98443 

44689 

98229 

47209 

98001 

45 

16 

42047 

98440 

44733 

98226 

47249 

97997 

44 

17 

42093 

98436 

44776 

98222 

47290 

97993 

43 

18 

42140 

98433 

44819 

98218 

47330 

97989 

42 

19 

42186 

98429 

44862 

98215 

47371 

97986 

41 

20 

9.42232 

9.98426 

9.44905 

9.98211 

9.47411 

9.97982 

40 

21 

42278 

98422 

44948 

98207 

47452 

97978 

39 

22 

42324 

98419 

44992 

98204 

47492 

97974 

38 

23 

42370 

98415 

45035 

98200 

47533 

97970 

37 

24 

42416 

98412 

45077 

98196 

47573 

97966 

36 

25 

42461 

98409 

45120 

98192 

47613 

97962 

35 

26 

42507 

98405 

45163 

98189 

47654 

97958 

34 

27 

42553 

98402 

45206 

98185 

47694 

97954 

33 

28 

42599 

9839S 

45249 

98181 

47734 

97950 

32 

29 

42644 

98395 

45292 

98177 

4)  t  li 

97946 

31 

30 

9.42690 

9.98391 

9.453.34 

9.98174 

9.47814 

9.97942 

30 

31 

42735 

98388 

45377 

98170 

47854 

97938 

29 

32 

42781 

98384 

45419 

98166 

47894 

97934 

28 

33 

42826 

98381 

45462 

98162 

47934 

97930 

27 

34 

42872 

98377 

45504 

98159 

47974 

97920 

26 

35 

42917 

98373 

45547 

98155 

48014 

97922 

25 

36 

42962 

98370 

45589 

98151 

48054 

97918 

24 

37 

43008 

98366 

45632 

98147 

48094 

97914 

23 

38 

43053 

98363 

45674 

98144 

48133 

97910 

22 

39 

43098 

98359 

45716 

98140 

48173 

97906 

21 

40 

9.43143 

9.98356 

9.45758 

9.98136 

9.48213 

9.97902 

20 

41 

43188 

98352 

45801 

98132 

48252 

97898 

^^ 

42 

43233 

98349 

45843 

98129 

48292 

97894 

18 

43 

43278 

98345 

45885 

98125 

48332 

97890 

17 

44 

43323 

98342 

45927 

98121 

4^371 

97886 

16 

45 

43367 

98338 

45969 

98117 

48411 

97882 

15 

46 

43412 

98334 

46011 

98113 

48450 

97878 

14 

47 

434.57 

98331 

46053 

98110 

48490 

97874 

13 

48 

43502 

98327 

46095 

98106 

48529 

97870 

12 

49 

43546 

98324 

46136 

98102 

48568 

97806 

11 

50 

9.43591 

9.98320 

9.46178 

9.98098 

9.48607 

9.97861 

10 

51 

43635 

98317 

46220 

98094 

48047 

97857 

9 

52 

43680 

98313 

46262 

98090 

48686 

97853 

8 

53 

43724 

98309 

46303 

98087 

48725 

97849 

1 

54 

43769 

98306 

46345 

98083 

48764 

97845 

6 

55 

43813 

98302 

46386 

98079 

48803 

97841 

5 

56 

43857 

98299 

46428 

98075 

48S42 

97a37 

4 

57 

43901 

98295 

46469 

98071 

48881 

97833 

3 

58 

43946 

98291 

46511 

98067 

48920 

97  829 

2 

59 

43990 

98288 

46552 

98063 

48959 

97825 

i 

60 

44034 

98284 

46594 

98060 

48998 

97821 

0 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

72° 

/ 

74° 

73° 

s 


^I^ZMgi^ 


■7      iT     ' 


244   TABLE  VIL— LOGARITHMIC   SIXES  AND  COSIXES. 


/ 

18° 

19° 

20° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.48998 

9.97821 

9.51264 

9.97567 

9.53405 

9.97299 

60 

1 

49037 

97817 

51301 

97563 

53440 

97294 

59 

2 

49076 

97812 

51338 

97558 

53475 

97289 

58 

3 

49115 

97808 

51374 

97554 

5:Bo09 

97285 

57 

4 

49153 

97804 

51411 

97550 

.5.3544 

97280 

50 

5 

49192 

97800 

51447 

97545 

53578 

97276 

55 

6 

49231 

97796 

51484 

97.541 

53613 

97271 

.54 

7 

49269 

97792 

51520 

97536 

53647 

97266 

53 

8 

49:i08 

97788 

51557 

97532 

53682 

97262 

52 

9 

49347 

97784 

51593 

97528 

53716 

97257 

51 

10 

9  49385 

9.97779 

9.51629 

9.97523 

9.. 53751 

9.97252 

50 

11 

49424 

97775 

51666 

97519 

53785 

97248 

49 

1-2 

49462 

97771 

51702 

97515 

5.3819 

97243 

48 

13 

49500 

97767 

51738 

97510 

53854 

97238 

47 

14 

49539 

97763 

51774 

97.506 

53888 

97234 

46 

15 

49577 

97759 

51811 

97501 

53922 

97229 

45 

16 

49615 

97754 

51847 

97497 

53957 

97224 

44 

17 

49654 

977.50 

518*3 

97492 

53991 

97220 

43 

18 

49692 

97746 

51919 

97488 

54025 

97215 

42 

19 

49730 

97742 

51955 

97484 

54059 

97210 

41 

20 

9.49768 

9.97738 

9.51091 

9.97479 

9.54093 

9.97206 

40 

21 

49806 

97734 

52027 

97475 

54127 

97201 

39 

22 

49844 

97729 

52063 

97470 

54161 

97196 

38 

23 

49882 

97725 

52099 

97466 

54195 

97192 

37 

24 

49920 

97721 

52135 

97461 

54229 

97187 

36 

25 

49958 

97717 

52171 

97457 

542^3 

97182 

:35 

26 

49996 

97713 

52207 

97453 

54297 

97178 

34 

27 

50034 

97708 

52242 

97448 

54331 

97173 

33 

28 

50072 

97704 

52278 

97444 

.54:365 

97168 

32 

29 

501 IQ 

97700 

52314 

97439 

54399 

97163 

31 

30 

9.50148 

9.97696 

9.52350 

9.97435 

9.54433 

9.97159 

30 

31 

50185 

97691 

52385 

97430 

54466 

971.54 

29 

32 

50223 

97687 

52421 

97426 

54500 

97149 

28 

33 

50261 

9768:3 

52456 

97421 

51.5.34 

97145 

27 

34 

50298 

97679 

52492 

97417 

54.507 

97140 

26 

35 

50336 

97674 

52527 

97412 

54601 

97135 

25 

36 

50374 

97670 

52563 

97408 

54635 

971.30 

24 

37 

50411 

97666 

52598 

97403 

54668 

97126 

23 

38 

50449 

97662 

52634 

97399 

54702 

97121 

22 

39 

50186 

97657 

52669 

97394 

547:35 

97116 

21 

40 

9.50523 

9.976.53 

9.52705 

9.97390 

9.54769 

9.97111 

20 

41 

.50501 

976i9 

52740 

97385 

54802 

97107 

19 

42 

5()59S 

97645 

52775 

97381 

548:36 

97102 

18 

43 

50635 

97640 

52811 

97376 

54869 

97097 

17 

44 

50(;73 

97636 

52846 

97372 

54903 

97092 

16 

45 

50710 

97632 

528S1 

97367 

54936 

97087 

15 

46 

50747 

97628 

52916 

97363 

54969 

970S3 

14 

47 

50784 

97623 

52951 

973.58 

55003 

97078 

13 

48 

50821 

97619 

52986 

973.53 

55036 

97073 

12 

49 

50858 

97615 

53021 

97349 

55069 

97068 

11 

50 

9.50S96 

9.97610 

9.53056 

9.97344 

9.55102 

9.97063 

10 

51 

5()9:'>3 

97606 

53092 

97340 

5.5136 

97059 

9 

5-i 

50970 

97602 

53126 

97335 

55169 

97054 

8 

53 

51007 

97597 

.53161 

97331 

55202 

97049 

r* 
i 

54 

51013 

97593 

53196 

97326 

55235 

97044 

6 

55 

51080 

97589 

53231 

97322 

55268 

97039 

5 

56  i 

51117 

97584 

53266 

97317 

55301 

97035 

4 

57 

51154 

97580 

53301 

97312 

55334 

97030 

3 

58  1 

51191 

97576 

53336 

97308 

55367 

97025 

0 

59 

51227 

97571 

53370 

97303 

55400 

97020 

1 

60 

51364 

97567 

53405 

97299 

55433 

97015 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

71° 

70° 

— ,^ — ^i^^ 

69° 

Q;ji 


1  /-^ 


nrtr 


111  11*>  11Q 


TABLE  YIT. 

— LOGA 

RITHMK 

r  SINES 

AND^C 

OSINES. 

Z^'O 

/ 

21° 

22 

o 

23 

o 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.55433 

9.97015 

9.573.58 

9.96717 

9.59188 

9.96403 

60 

1 

55466 

97010 

.57389 

96711 

59218 

96397 

59 

o 

55499 

97005 

57420 

96706 

59247 

96392 

58 

3 

55532 

97001 

57451 

96701 

59277 

963-S7 

57 

4 

55564 

96996 

57482 

96696 

59307 

96381 

56 

5 

55597 

96991 

57514 

96691 

59336 

96376 

55 

6 

55630 

96986 

57545 

96686 

59.366 

96:^70 

54 

7 

55663 

96981 

57576 

96681 

59396 

9(365 

53 

8 

55695 

96976 

57607 

96676 

59425 

96360 

52 

9 

55728 

96971 

57638 

96670 

59455 

9C354 

51 

10 

9.55761 

9.96966 

9.57669 

9.96665 

9.59484 

9.96349 

50 

11 

55793 

90962 

57700 

96660 

59514 

96343 

49 

12 

55826 

96957 

57731 

96655 

59543 

96388 

48 

13 

55858 

96952 

57762 

96650 

59573 

96333 

47 

14 

55891 

96947 

57793 

96645 

59602 

96327 

46 

15 

55923 

96942 

57824 

96640 

59632 

96322 

45 

16 

55956 

96937 

57855 

96634 

59661 

96316 

44 

17 

55988 

96932 

57885 

96629 

59690 

96311  1 

43 

18 

56021 

96927 

57916 

96624 

59720 

96305  i 

42 

19 

56053 

96922 

.57947 

96619 

59749 

96300 

41 

20 

9.56085 

9.96917 

9.57978 

9.96614 

9.. 59778 

9.96294 

40 

21 

56118 

96912 

58008 

96608 

59808 

96289 

39 

22 

56150 

9()907 

58039 

96603 

59837 

96284  1 

38 

23 

56182 

96903 

58070 

96598 

59866 

96278 

37 

24 

56215 

96898 

58101 

96593 

59895 

96273 

36 

25 

56247 

96893 

58131 

96588 

59924 

96267 

35 

26 

56279 

96888 

58162 

96582 

599.54 

96262 

34 

27 

56311 

96883 

58192 

96577 

59983 

96256 

33 

28 

56343 

96878 

58223 

96572 

60012 

96-J51 

32 

29 

56375 

96873 

58253 

96567 

60041 

96245 

31 

30 

9.56408 

9.96868 

9.58284 

9.96562 

9.60070 

9.96240 

30 

31 

56440 

96863 

58314 

96556 

60099 

96234 

29 

32 

56472 

96858 

58345 

96,551 

60128 

96229 

28 

33 

56504 

96853 

58375 

96546 

60157 

962V3 

27 

34 

56536 

96848 

58406 

96541 

60186 

9*218 

26 

35 

56568 

96843 

58436 

96535 

60215 

96212 

25 

36 

56599 

96838 

58467 

96530 

60244 

96207 

24 

37 

56631 

96833 

58497 

96525 

60273 

96201 

23 

38 

56663 

96828 

58527 

96520 

60302 

96196 

22 

39 

56695 

96823 

58557 

96514 

60331 

96190 

21 

40 

9.56727 

9.96818 

9.-58588 

9.96509 

9.60359 

9.96185 

20 

41 

56759 

96813 

.58618 

96504 

60:^88 

96179 

19 

42 

56790 

96808 

58648 

96498 

60417 

96174 

18 

43 

56822 

96803 

58678 

96493 

60446 

96168 

17 

44 

568.54 

96798 

58709 

96488 

60474 

96162 

16 

45 

56886 

96793 

58739 

96483 

60503 

96157 

15 

46 

.56917 

96788 

58769 

96477 

60532 

96151 

14 

47 

56949 

96783 

58799 

96472 

60561 

96146 

13 

48 

56980 

96778 

58829 

96467 

60589 

96140 

12 

49 

57012 

96772 

58859 

96461 

60618 

96135 

11 

50 

9.57044 

9.96767 

9.58889 

9.96456 

9.60646 

9.96129 

10 

51 

.57075 

96762 

58919 

96451 

60<;75 

96123 

9 

52 

57107 

967.57 

58949 

96445 

60704 

96118 

8 

53 

57138 

96752 

58979 

96440 

60732 

96112 

7 

54 

57169 

96747 

59009 

96435 

60761 

96107 

6 

55 

57201 

96742 

59039 

96429 

60789 

96101 

5 

56 

57232 

96737 

59069 

96424 

60818 

96095 

4 

57 

57264 

96732 

59098 

96419 

60846 

96090 

3 

58 

57295 

96727 

.59128 

96413 

60875 

96084 

2 

59 

57326 

96722 

59158 

96408 

60903 

96079 

1 

60 
/ 

57.358 

96717 

59188 

96403 

60931 

96073 

0 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

68° 

67° 

66° 

;W?5e 

I.* 

246   TABLE  W.— LOG ARli'ltflC   SINES   In'd'' COSINES. 


/ 

24° 

25° 

26° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Co.sine 

0 

9.60931 

9.96073 

9.62595 

9.95728 

9.64184 

9.95366 

60 

1 

G0960 

96067 

62622 

95722 

64:il0 

95360 

59 

2 

60988 

96062 

62649 

95716 

64236 

95354 

58 

3 

61016 

96056 

62676 

95710 

64262 

95348 

57 

4 

61045 

96050 

62703 

95704 

64288 

95341 

56 

5 

61073 

96045 

62730 

95698 

64313 

95335 

55 

6 

61101 

96039 

62757 

95692 

64339 

95329 

54 

61129 

96034 

62784 

95686 

64365 

95323 

53 

8 

61158 

96028 

62811 

95680 

64391 

95317 

52 

9 

61186 

96022 

62838 

95674 

64417 

95310 

51 

10 

9.61214 

9.96017 

9.62865 

9.95668 

9.64442 

9.95304 

50 

11 

61242 

96011 

62892 

95663 

64468 

95298 

49 

12 

61270 

96005 

62918 

95657 

64494 

95292 

48 

13 

61298 

96000 

62945 

95651 

64519 

95286 

47 

14 

61326 

95994 

62972 

95645 

64545 

95279 

46 

15 

61354 

95988 

62999 

95639 

64571 

95273 

45 

16 

61382 

95982 

63026 

95633 

64596 

95267 

44 

17 

61411 

95977 

63052 

95627 

64622 

95201 

43 

18 

61438 

95971 

63079 

95621 

64647 

95254 

42 

19 

61466 

95965 

63106 

95615 

64673 

95248 

41 

20 

9.61494 

9.95960 

9.63i:33 

9.95609 

9.64698 

9.95242 

40 

21 

6152i 

95954 

63159 

95603 

64724 

95236 

39 

22 

61550 

95918 

63186 

95597 

64749 

95229 

38 

23 

61578 

95942 

63213 

95.591 

64775 

95223 

37 

24 

61606 

93937 

63239 

95585 

64800 

95217 

36 

25 

61631 

95931 

63266 

95579 

64826 

95211 

35 

26 

61662 

95925 

63292 

95573 

64851 

95204 

34 

27 

61689 

95920 

63319 

95.567 

64877 

95198 

33 

28 

61717 

95914 

63345 

95561 

64902 

95192 

32 

29 

61745 

95908 

63372 

95555 

64927 

95185 

31 

30 

9.61773 

9.95902 

9.63398 

9.95549 

9.64953 

9.95179 

30 

31 

61800 

95897 

63425 

93.543 

64978 

95173 

29 

32 

61828 

95891 

63451 

95537 

65003 

95167 

28 

33 

61856 

958S5 

63478 

95531 

65029 

95160 

27 

34 

61883 

95879 

C3504 

95525 

65054 

95154 

26 

35 

61911 

95873 

63531 

95519 

65079 

95148 

25 

36 

61939 

95S68 

63557 

95513 

65104 

93141 

24 

37 

61966 

95862 

63583 

95507 

65130 

95135 

23 

38 

61994 

95836 

63610 

93500 

63155 

95129 

22 

39 

62021 

95850 

63636 

95494 

65180 

95122 

21 

40 

9.62049 

9.95844 

9.63662 

9.95488 

9.65205 

9.95116 

20 

41 

62076 

958C9 

63689 

95482 

65230 

95110 

19 

42 

62104 

95833 

63715 

95476 

65255 

9.3103 

18 

43 

62131 

95827 

63741 

95470 

65281 

95097 

17 

44 

62159 

9.5821 

63767 

95464 

65306 

95090 

16 

45 

62186 

95815 

63794 

95458 

63331 

95084 

15 

46 

62214 

95810 

63820 

95452 

65356 

95078 

14 

47 

62241 

95804 

63846 

95446 

65381 

95071 

13 

48 

62268 

95798 

63S72 

95440 

65406 

93065 

12 

49 

62296 

95792 

63898 

95434 

65431 

95059 

11 

■  50 

9.62323 

9.95786 

9.63924 

9.95427 

9.6.5456 

9.95052 

10 

51 

62350 

95780 

63950 

95421 

65481 

95046 

9 

52 

62377 

95775 

63976 

95415 

65508 

95039 

8 

53 

62405 

95769 

64002 

95409 

65531 

93033 

7 

54 

62432 

95763 

64028 

95403 

65565 

95027 

6 

55 

62459 

90(0. 

04054 

95397 

65580 

95020 

5 

56 

62486 

95751 

640S0 

9.3391 

65605 

95014 

4 

57 

62513 

95745 

64106 

95384 

65630 

95007 

3 

58 

62541 

95739 

64132 

95378 

65655 

95001 

2 

59 

62568 

93733 

64158 

95372 

65680 

94995 

1 

60 

62595 

95728 

64184 

95366 

65705 

949R8 

0 

/. 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

' 

65° 

64° 

63° 

1 

—  r— 

^  '"  /< 

1 

si^. 

<-^^^. 

»  AA 

i.^'i*. 

^    c^ 

'.  Cw 

hm 

ttfVtti 

9iSi 

fi,€c 

s. 

CQtf.Stii.Cos.  Sin. Cos.  Sin. 

n-iH-  ""IR  1iO 

TABLE  VII'.--LOGARITH^iI(;  'SINES  AND  COSINES.    24/ 


/ 

2 

«" 

28' 

* 

29° 

/ 

Sine 

Cosine 
9.94988 

Sine 

Cosine 

Sine 

Cosine 

0 

9.65705 

9.67161 

9.94.593 

9.68557 

9.94182 

60 

1 

65729 

94982 

67186 

94587 

68580 

94175 

59 

2 

65754 

94975 

67208 

94580 

68603 

94168 

58 

3 

65779 

94909 

67232 

94573 

68025 

94161 

57 

4 

65804 

94962 

67256 

94567 

68648 

94154 

56 

5 

65828 

94956 

67280 

94560 

68071 

94147 

55 

6 

65853 

94949 

67303 

94553 

68694 

94140 

54 

i 

65878 

94943 

67327 

94546 

68716 

94133 

53 

8 

65902 

94936 

67350 

94540 

68739 

94126 

52 

9 

65927 

94930 

67374 

94533 

68762 

94119 

51 

10 

9.65952 

9.94923 

9.67398 

9.94526 

9.68784 

9.94112 

50 

11 

65976 

94917 

67421 

94519 

68807 

94105 

49 

12 

66001 

94911 

67445 

94513 

68829 

94098 

48 

i3 

66025 

94904 

67468 

94506 

68852 

94090 

47 

14 

66050 

94898 

67492 

94499 

6S875 

94083 

46 

15 

66075 

94891 

67515 

94492 

68897 

94076 

45 

16 

66099 

94885 

67539 

94485 

68920 

94009 

44 

17 

66124 

94878 

67.562 

94479 

68942 

94062 

43 

18 

66148 

94871 

67586 

94472 

68965 

94055 

42 

19 

66173 

94865 

67609 

94465 

68987 

94048 

41 

20 

9.66197 

9.94858 

9.67633 

9.94458 

9.69010 

9.94041 

40 

21 

66221 

948.52 

676.56 

94451 

69032 

94034 

39 

22 

66246 

94845 

67680 

94445 

69055 

94027 

38 

23 

662';0 

94839 

67703 

94438 

69077 

94020 

37 

24 

66295 

9483-2 

67726 

94431 

69100 

94012 

36 

25 

66319 

94826 

67750 

94424 

69122 

94005 

35 

26 

66343 

94819 

67773 

94417 

69144 

93998 

34 

27 
28 

60368 
66392 

94813 
94806 

67796 

67820 

94410 
94404 

69167 
69189 

93991 
93984 

33 
32 

29 

66416 

94799 

67843 

94397 

69212 

93977 

31 

30 

9.66441 

9.94793 

9.67866 

9.94390 

9.69234 

9.93970 

30 

31 

66165 

94786 

67890 

94383 

69256 

93963 

29 

32 

66489 

94780 

67913 

94376 

69279 

93955 

28 

33 

66513 

94773 

67936 

94369 

69301 

93948 

2i 

34 

66537 

94767 

67959 

94362 

69323 

93941 

26 

35 

66502 

94700 

67982 

94355 

69345 

93934 

25 

36 

665^6 

94753 

68006 

94349 

69368 

93927 

24 

37 

60610 

94747 

08029 

94342 

69390 

93920 

23 

38 

66634 

94740 

08052 

94335 

69412 

93912 

22 

39 

66658 

94734 

68075 

94328 

69434 

93905 

21 

40 

9.66682 

9.94727 

9.68098 

9.94321 

9.69456 

9.93898 

20 

41 

66706 

94720 

68121 

94314 

69479 

93891 

19 

1  o 

42 

66731 

94714 

68144 

94307 

69.501 

93884 

18 

4  r* 

43 

60755 

94707 

68167 

94300 

69523 

93876 

17 

1  iS 

44 

66779 

94';  00 

68190 

94293 

69545 

93869 

lo 

45 

00803 

94694 

68213 

94286 

69567 

93862 

15 

40 

60827 

94687 

68237 

94279 

69589 

93855 

14 

-1  O 

47 

66851 

94080 

68260 

94273 

69611 

93847 

13 

1  O 

48 

60875 

94674 

08283 

94266 

69638 

93840 

12 

49 

60899 

94067 

68305 

94259 

69655 

93833 

11 

50 

9.66922 

9.94660 

9.68328 

9.94252 

9.69677 

9.93826 

10  ' 
9 

8 

51 

6C946 

94654 

68351 

94245 

69699 

93819 

52 

00970 

94647 

68374 

94238 

69721 

93811 

53 

60994 

94640 

68397 

94231 

69743 

93804 

4 

6 
5 

54 

67018 

94634 

68420 

94224 

69765 

93797 

55 

67042 

94627 

68443 

94217 

69787 

93789 

56 

67066 

94620 

68466 

94210 

69809 

93782 

4 
3 
2 

57 

67090 

94614 

68489 

94203 

69831 

93775 

58 

67113 

94607 

68512 

94196 

69853 

93768 

59 

67137 

94000 

68534 

94189 

69875 

93760 

1 

60 

67101 

94593 

68557 

94182 

69897 

93753 

0 

/ 

Cosine 

Sine 
62° 

Cosine 

Sine 

Cosine 

Sine 

/ 

61° 

60° 

"7 

%•  'i  -1- 

■). 

.!< 

W" 

o%f-. 

>  ;•  .  ; 

1 « 

n  ^O.^^Xi    ■ 

-J 

on 

t 

2] 

-»  o  o 

248 

table'  V!t.— logarithmic  sines  ano  rosiNEs. 

t 

30° 

31° 

32° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine    Cosine 

0 

9.69897 

9.93753 

9.71184 

9.93307 

9.72421    9.92842 

60 

1 

69919 

93746 

71205 

93299 

72441     92834 

59 

2 

69941 

93738 

71226 

93291 

72461     92826 

58 

3 

69963 

93731 

71247 

93284 

72482     92818 

57 

4 

69984 

93724 

71268 

93276 

72502     92810 

56 

5 

70006 

93717 

71289 

93269 

72522     92803 

55 

6 

70028 

93709 

71.310 

93261 

72542     92795 

54 

7 

70050 

93702 

71331 

93253 

72562     92787 

53 

8 

70072 

93595 

71352 

93246 

72582     92779 

52 

9 

70093 

93687 

71373 

93238 

72602     92771 

51 

10 

9.70115 

9.93680 

9.71393 

9.93230 

9.72622   9.92763 

50 

11 

70137 

93673 

71414 

932-,'3 

72043     92755 

49 

12 

70159 

93665 

71435 

93215 

72663     92747 

48 

13 

70180 

93658 

71456 

93207 

72683     92739 

47 

14 

70202 

93650 

71477 

93200 

72703     92731 

46 

15 

70224 

93643 

71498 

93192 

72723     92723 

45 

16 

70245 

93636 

71519 

93184 

72743     92715 

44 

17 

70267 

93628 

71539 

93177 

72763     92707 

43 

18 

70288 

93621 

71560 

93169 

72783     92C99 

42 

19 

70310 

93614 

71581 

93161 

72803     92691 

41 

20 

9.70332 

9.93606 

9.71602 

9.93154 

9.72823   9.92683 

40 

21 

70353 

93599 

71622 

93146 

72843     92675 

39 

22 

70375 

93591 

71643 

93138 

72863     92667 

38 

23 

70396 

93584 

71664 

93131 

72883     92659 

37 

24 

70418 

93577 

71685 

93123 

72902     92651 

o  1 

36 

25 

70439 

93569 

71705 

93115 

72922     92643 

35 

26 

70461 

93562 

71726 

93108 

72942     92635 

34 

27 

70482 

93554 

71747 

93100 

72962     92627 

33 

28 

70504 

93547 

71767 

93092 

72982     92619 

32 

29 

70525 

93539 

71788 

93084 

73002     92611 

31 

30 

9.70547 

9.93532 

9.71809 

9.93077 

9.73022   9.92603 

30 

31 

70568 

93525 

71829 

93069 

73041     92595 

29 

32 

70590 

93517 

71850 

93061 

73061     92587 

28 

33 

70611 

93510 

71870 

9.3053 

73081     92579 

27 

34 

70633 

93502 

71891 

93046 

73101     92571 

26 

35 

70654 

93495 

71911 

93038 

73121     92563 

25 

36 

70675 

93487 

71932 

93030 

73140     92555 

24 

37 

70697 

93480 

71952 

93022 

73160     92546 

23 

38 

70718 

93472 

71973 

93014 

73180     92538 

22 

39 

70739 

93465 

71994 

93007 

73200     92530 

21 

40 

9.70761 

9.93457 

9.72014 

9.92999 

9.73219   9.92522 

20 

41 

70782 

93450 

72034 

92991 

732.39     92514 

19 

42 

70803 

93442 

72055 

92983 

73259     92506 

18 

43 

70824 

934:35 

72075 

92976 

73278     92498 

17 

44 

70846- 

93427 

72096 

92968 

73298     92490 

16 

45 

70867 

93420 

72116 

92960 

73318     92482 

15 

46 

70888 

93412 

72137 

92952 

73337     92473 

14 

47 

70909 

93405 

72157 

92944 

73357     92465 

13 

48 

70931 

93397 

72177 

92936 

73377     92457 

12 

49 

70952 

93390 

72198 

92929 

73396     92449 

11 

50 

9.70978 

9.93382 

9.72218 

9.92921 

9.73416   9.92441 

10 

51 

70994 

93375 

72238 

92913 

73435     92433 

9 

52 

71015 

93367 

72259 

92905 

73455     92425 

8 

53 

71036 

93360 

72279 

92897 

73474     92416 

54 

71058 

93.352 

72299 

92889 

73494     92408 

6 

55 

71079 

93344 

72320 

92881 

73513     92400 

5 

56 

71100 

93337 

72340 

92874 

735.33     92392 

4 

57 

71121 

93329 

72360 

92866 

73552     92384 

3 

58 

71142 

93322 

72881 

92858 

73572     92376 

o 

59 

71163 

93314 

72401 

92850 

73591     92367 

1 

60 

71184 

93307 

72421 

92842 

73611     92359 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine    Sine 

^  / 

69° 

«8° 

57° 

♦  A 

Slnt 

Sih 

1^ 

.■jSIj^:Pf>y. 

Uos.tSiu.ios.  Sm,Vofi.  »m. 

-IQO  ^*^/l  "If)!- 

TABT.F^rIr— LOGARITkMie   SINES  Al^lTcfcsINES.   24D 


1 

33° 

34° 

85° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.73611 

9.92359 

9.74756 

9.918.57 

9.758.59 

9.91336 

60 

1 

73630 

92351 

74775 

91849 

75877 

91328 

59 

2 

73650 

92343 

74794 

91840 

75895 

91319 

58 

3 

73669 

92335 

74812 

91832 

75913 

91310 

57 

4 

73689 

92326 

74831 

91823 

75931 

91301 

56 

5 

73708 

92318 

74850 

91815 

7.5949 

91292 

55 

6 

73727 

92310 

74868 

91806 

75967 

91283 

54 

r* 
f 

73747 

92302 

74887 

91798 

75985 

91274 

53 

8 

73766 

92293 

74906 

91789 

76003 

91266 

52 

9 

73785 

92285 

74924 

91781 

76021 

91257 

51 

10 

9.73805 

9.92277 

9.74943 

9.91772 

9.76039 

9.91248 

50 

11 

73824 

92269 

74961 

91763 

76057 

91239 

49 

12 

73813 

922C0 

74980 

91755 

76075 

91230 

48 

13 

73863 

92252 

74999 

91746 

76093 

91221 

47 

14 

73882 

92244 

75017 

91738 

76111 

91212 

46 

15 

73901 

92235 

75036 

91729 

76129 

91203 

45 

16 

73921 

92227 

75054 

91720 

76146 

91194 

44 

17 

73940 

92219 

75073 

91712 

76164 

91185 

43 

18 

73959 

92211 

75091 

91703 

76182 

91176 

42 

19 

73978 

92202 

75110 

91695 

76200 

91167 

41 

20 

9.73997 

9.92194 

9.75128 

9. 91  (-86 

9.76218 

9.91158 

40 

21 

74017 

92186 

75147 

91677 

76236 

91149 

39 

22 

74036 

92177 

75165 

91669 

76253 

91141 

38 

28 

74055 

92169 

75184 

91660 

76271 

91132 

37 

24 

74074 

92161 

75202 

91651 

76289 

91123 

36 

25 

74093 

92152 

75221 

91643 

76307 

91114 

35 

26 

74113 

92144 

75239 

91634 

76324 

91105 

34 

27 

74132 

92136 

75258 

91625 

76342 

91096 

33 

28 

74151 

92127 

75276 

91617 

76360 

91087 

32 

29 

74170 

92119 

75294 

91608 

76378 

91078 

31 

30 

9.74189 

9.92111 

9.75313 

9.91599 

9.76395 

9.91069 

30 

31 

74208 

92102 

75331 

91591 

76413 

91060 

29 

32 

74227 

92094 

75350 

91582 

76431 

91051 

28 

33 

74246 

92086 

75368 

91573 

76448 

91042 

27 

34 

'<4265 

92077 

75386 

91565 

76466 

91033 

26 

35 

74284 

92069 

75405 

91556 

76484 

91023 

25 

36 

74303 

92060 

75423 

91547 

76501 

91014 

24 

37 

74322 

92052 

75441 

91538 

76519 

91005 

23 

38 

74341 

92044 

75459 

91530 

76537 

90996 

22 

39 

74360 

92035 

75478 

91521 

76554 

90987 

21 

40 

9.74379 

9.92027 

9.75496 

9.91512 

9.76572 

9.90978 

20 

41 

74398 

92018 

75514 

91504 

76590 

90969 

19 

42 

74417 

92010 

75533 

91495 

76607 

90960 

18 

43 

74436 

92002 

75551 

91486 

76625 

90951 

17 

44 

74455 

91993 

75569 

91477 

76642 

90942 

16 

45 

74474 

91985 

7.5587 

91469 

76660 

90933 

15 

46 

74493 

91976 

75605 

91460 

76677 

90924 

14 

47 

74512 

91968 

75624 

91451 

76695 

90915 

13 

48 

74531 

91959 

75642 

91442 

76712 

90906 

12 

49 

74549 

91951 

75660 

91433 

76730 

90896 

11 

50 

9.74568 

9.91942 

9.75678 

9.91425 

9.76747 

9.90887 

10 

51 

74587 

91934 

75696 

91416 

76765 

90878 

9 

52 

74606 

91925 

7.5714 

91407 

76782 

90869 

8 

53 

74625 

91917 

75733 

91398 

76800 

90860 

1 

54 

74644 

91908 

7.5751 

91.389 

76817 

90851 

6 

55 

74662 

91900 

75769 

91381 

76835 

90842 

5 

56 

74681 

91891 

75787 

91372 

76852 

90832 

4 

57 

74700 

91883 

75805 

91363 

76870 

90823 

3 

58 

74719 

91874 

75823 

91354 

76887 

90814 

2 

59 

74737 

91866 

75841 

91345 

76904 

90805 

1 

60 

74756 

91857 

75859 

91336 

76922 

90796 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

66» 

65° 

54» 

•1 

-jC^ 

f 

250  TABLE^ti.— LOGARITHMIC   SINES  'aS0  COSINES. 


0 

36° 

37° 

1 

4 

J8° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.76922 

9.90796 

9.77946 

9.90235  ■ 

9.78934 

9.89653 

60 

1 

76939 

90787 

77963 

90225 

78950 

89643 

59 

2 

76957 

90777 

77980 

90216 

78967 

89633 

58 

3 

76974 

90768 

77997 

90206 

78983 

89624 

57 

4 

76991 

90759 

78013 

90197 

78999 

89614 

56 

5 

77009 

90750 

78030 

90187 

79015 

89604 

55 

6 

77026 

90741 

78047 

90178 

79031 

89594 

54 

7 

77043 

90731 

78063 

90168 

79047 

89584 

53 

8 

77061 

90722 

78080 

90159 

79063 

89574 

52 

9 

77078 

90713 

78097 

90149 

79079 

89564 

51 

10 

9.77095 

9.90704 

9.78113 

9.90139 

9.79095 

9.89554 

50 

11 

77112 

90694 

78130 

90130 

79111 

89544 

49 

12 

77130 

90685 

78147 

90120 

79128 

89534 

48 

13 

77147 

90676 

78163 

90111 

79144 

89524 

47 

14 

77164 

90667 

78180 

90101 

79160 

89514 

46 

15 

77181 

90657 

78197 

90091 

79176 

89504 

45 

16 

77199 

90648 

78213 

90082 

79192 

89495 

44 

17 

77216 

90639 

78230 

90072 

79208 

89485 

43 

18 

77233 

90630 

78246 

90063 

79224 

89475 

42 

19 

77250 

90620 

78263 

90053 

79240 

89465 

41 

20 

9.77268 

9.90611 

9.78280 

9.90043 

9.79256 

9.89455 

40 

21 

77285 

90602 

78296 

90034 

79272 

89445 

39 

22 

77302 

90592 

78313 

90024 

79288 

89435 

38 

23 

77319 

90583 

78329 

90014 

79304 

89425 

37 

24 

77336 

90574 

7834G 

90005 

79319 

89415 

36 

25 

77353 

90565 

78362 

89995 

79335 

89405 

35 

26 

77370 

90555 

78379 

89985 

79351 

89395 

34 

27 

77387 

90546 

78395 

89976 

79367 

89385 

33 

28 

77405 

90537 

78412 

89966 

79383 

89375 

32 

29 

77422 

90527 

78428 

89956 

79399 

89364 

31 

30 

9.77439 

9.90518 

9.78445 

9.89947 

9.79415 

9.89354 

30 

31 

77456 

90509 

78461 

89937 

79431 

89344 

29 

32 

77473 

90499 

78478 

89927 

79447 

H9334 

28 

33 

77490 

90490 

78494 

89918 

79463 

89324 

27 

34 

77507 

90480 

78510 

89908 

79478 

89314 

26 

35 

77524 

90471 

78527 

89898 

79494 

89304 

25 

36 

77541 

90462 

78543 

89888 

79510 

89294 

24 

37 

77558 

90452 

78560 

89879 

79.526 

89284 

23 

38 

77575 

90443 

78576 

89869 

79542 

89274 

22 

39 

77592 

90434 

78592 

89859 

79558 

89264 

21 

40 

9.77609 

9.90424 

9.78609 

9.89849 

9.79573 

9.89254 

20 

41 

77626 

90415 

78625 

89840 

79589 

89244 

19 

42 

77643 

90405 

78642 

89830 

79605 

89233 

18 

43 

77660 

90396 

7S658 

89820 

78621 

89223 

17 

44 

77677 

90386 

78674 

80810 

7963(; 

89213 

16 

45 

77694 

90377 

78691 

89801 

79652 

89203 

15 

46 

77711 

9036S 

78707 

89791 

79668 

89193 

14 

47 

77728 

90358 

78723 

89781 

79684 

89183 

13 

48 

77744 

90349 

78739 

89771 

79699 

89173 

12 

49 

77761 

90339 

78756 

89761 

79715 

89162 

11 

50 

9.77778 

9.90330 

9.78772 

9.89752 

9.79731 

9.89152 

10 

51 

77795 

90320 

78788 

89742 

79746 

89142 

9 

52 

77812 

90311 

78805 

89732 

79762 

89132 

8 

53 

77829 

90301 

78821 

89722 

79778 

89122 

7 

54 

77846 

90292 

78837 

89712 

79793 

89112 

6 

55 

77862 

90282 

78853 

89702 

79809 

89101 

5 

56 

77879 

90273 

78869 

89693 

79825 

89091 

4 

57 

77896 

90263 

78886 

89683 

79840 

89081 

3 

58 

77913 

90254 

78902 

89673 

79856 

89071 

2 

59 

77930 

90244 

78918 

89663 

79872 

89060 

1 

60 

77946 

90235 

78934 

89653 

79887 

89050 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

i 

53° 

52° 

51° 

^;:k%5.  sM^s.  sin}cas. 


C^^^.Sm.  C(^^^  Sin.  Cqh,,  Sih, 

TABLE  Vtt.—LOGARITlilirtC  SINES  ANli  COSINES.   251 


/ 

89° 

40° 

41° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.79887 

9.89050 

9.80807 

9.88425 

9.81694 

9.87778 

60 

1 

79903 

89040 

80822 

88415 

81709 

87767 

59 

2 

79918 

89030 

80837 

88404 

81723 

87756 

58 

3 

79934 

89020 

80852 

88394 

81738 

87745 

57 

4 

79950 

89009 

80867 

88383 

81752 

87734 

56 

5 

79965 

88999 

80882 

88372 

81767 

87723 

55 

6 

79981 

88989 

80897 

88362 

81781 

87712 

54 

< 

79996 

88978 

80912 

88351 

81796 

87701 

53 

8 

80012 

88968 

80927 

88340 

81810 

87690 

52 

9 

80027 

88958 

80942 

88330 

81825 

87679 

51 

1 

10 

9.80043 

9.88948 

9.80957 

9.88319 

9.81839 

9.87668 

50 

11 

80058 

88937 

80972 

88308 

81854 

87657 

'  49 

1~' 

80074 

88927 

80987 

88298 

81868 

87646 

48 

13 

80089 

88917 

81002 

88287 

81882 

87635 

47 

14 

80105 

88906 

81017 

88276 

81897 

87624 

46 

15 

80120 

88896 

81032 

88266 

81911 

87613 

45 

16 

80136 

88886 

81047 

88255 

81926 

87601 

44 

17 

80151 

88875 

81061 

88244 

81940 

87590 

43 

18 

80166 

88865 

81076 

88234 

819.55 

87579 

42 

19 

80182 

88855 

81091 

88223 

81969 

87568 

41 

20 

9.80197 

9.88844 

9.81106 

9.88212 

9.81983 

9.87557 

40 

^1 

80213 

88834 

81121 

88201 

81998 

87546 

39 

8'> 

80228 

8S824 

81136 

88191 

82012 

87535 

38 

2:i 

80244 

88813 

81151 

88180 

82026 

87524 

37 

24 

80259 

8S803 

81166 

88169 

82041 

87513 

36 

25 

80274 

88793 

81180 

88158 

820.')5 

87501 

35 

26 

80290 

88782 

81195 

88148 

82069 

87490 

34 

27 

80305 

88772 

81210 

88137 

82084 

87479 

33 

28 

80320 

88761 

81225 

88d26 

82098 

87468 

32 

29 

80336 

88751 

81240 

88115 

82112 

87457 

31 

30 

9.80351 

9.88741 

9.81254 

9.88105 

9.82126 

9.87446 

30 

31 

80866 

88730 

81269 

88094 

82141 

87434 

29 

32 

80382 

88720 

81284 

88083 

82155 

87423 

28 

33 

80397 

88709 

81299 

88072 

82169 

87412 

27 

34 

80412 

88699 

81314 

88061 

82184 

87401 

26 

35 

80428 

88688 

81328 

88051 

82198 

87390 

25 

36 

80443 

88678 

81343 

88040 

82212 

87378 

24 

37 

80458 

88668 

81358 

88029 

82226 

87367 

23 

38 

80473 

88657 

81372 

88018 

82240 

87356 

22 

39 

80489 

88647 

81387 

88007 

82255 

87345 

21 

40 

9.80504 

9.88636 

9.81402 

9.87996 

9.82269 

9.87334 

20 

41 

80519 

88626 

81417 

87985 

82283 

87322 

19 

42 

80534 

88615 

81431 

87975 

82297 

87311 

18 

43 

80550 

88605 

81446 

87964 

82311 

87300 

17 

44 

80565 

88594 

81461 

87953 

82326 

87288 

16 

45 

80580 

88584 

81475 

87942 

82340 

87277 

15 

46 

80595 

8^^573 

81490 

87931 

82354 

87266 

14 

47 

80610 

88563 

81505 

87920 

82368 

87255 

13 

48 

80625 

88552 

81.519 

87909 

82382 

87243 

12 

49 

80641 

88542 

81534 

87898 

82396 

87232 

11 

50 

9.80656 

9.88531 

9.81549 

9.87887 

9.82410 

9.87221 

10 

51 

80671 

88521 

81563 

87877 

82424 

87209 

9 

52 

80686 

88510 

81578 

87866 

82439 

87198 

8 

53 

80701 

88499 

81592 

87855 

824.^3 

87187 

7 

54 

80716 

88489 

81607 

87844 

82467 

87175 

6 

55 

80731 

88478 

81622 

87833 

82481 

87164 

5 

56 

80746 

88468 

81036 

87822 

82495 

87153 

4 

57 

80762 

88457 

81651 

87811 

82509 

87141 

3 

58 

80777 

88147 

81665 

87800 

82523 

87130 

2 

59 

80792 

88436 

81680 

87789 

82537 

87119 

1 

60 

80807 

88425 

81694 

87778 

82551 

87107 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

60° 

49° 

48° 

3>^; 


Sii'ifCos.  sin -Cos.  SmlCat. 


m 


TABLE 

Xm.  LC 

)GARIT 

HMrC  SI 

NES  A! 

^D'COSIi 

STES. 

1 

42° 

43° 

44° 

/ 



1   Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.82551 

9.8710? 

9.83378 

9.86413 

9.84177 

9.85693 

60 

1 

!   8:2565 

87096 

8:3392 

86401 

84190 

85681 

59 

o 

82579 

87085 

83405 

86389 

84203 

85669 

58 

3 

82593 

87073 

83419 

86377 

84216 

85657 

57 

4 

82607 

87062 

83432 

86366 

84229 

85645 

56 

5 

82621 

87050 

8:3446 

86354 

84242 

85632 

55 

6 

82635 

87039 

83459 

86:342 

84255 

85020 

54 

7 

82649 

87028 

83473 

86:330 

84269 

85608 

53 

8 

82663 

87016 

83486 

86318 

84282 

85596 

52 

9 

82677 

87005 

83500 

86306 

84295 

85583 

51 

10 

9.82691 

9.86993 

9.83513 

9.86295 

9.84308 

9.85571 

50 

11 

82T05 

86982 

83527 

86283 

84321 

85559 

49 

12 

82719 

86970 

83540 

86271 

84334 

85547 

48 

13 

82733 

86959 

83554 

86259 

84347 

85534 

47 

14 

82747 

86947 

8:3567 

86247 

84360 

85522 

46 

15 

82761 

86936 

83581 

86235 

84373 

85510 

45 

16 

82775 

86924 

835^ 

86223 

84385 

85497 

44 

17 

82788 

86913 

83608 

86211 

84398 

8.5485 

43 

18 

82802 

86902 

83621 

86200 

84411 

85473 

42 

19 

82816 

86890 

83634 

86188 

84424 

85460 

41 

20 

-  9.82830 

9.86879 

9.a3648 

9.86176 

9.84437 

9.8.5448 

40 

21 

82844 

86867 

83661 

86164 

84450 

85436 

39 

22 

82858 

86855 

83674 

86152 

84463 

85423 

38 

23 

82872 

86844 

83688 

86140 

84476 

8.5411 

37 

24 

82885 

86a32 

83701 

86128 

84489 

85399 

36 

25 

82899 

86821 

8:3715 

86116 

84502 

85386 

35 

26 

82913 

86809 

83728 

86104 

84515 

85374 

34 

27 

82927 

86798 

83741 

86092 

84528 

85361 

33 

28 

82941 

86786 

83755 

86080 

84540 

85349 

32 

29 

82955 

86775 

83768 

86068 

84553 

85337 

31 

30 

9.82968 

9.86763 

9.83781 

9.86056 

9.84566 

9.85324 

;30 

31 

82982 

86752 

83795 

86044 

84579 

85312 

29 

32 

82996 

86740 

8:3808 

86032 

84592 

85299 

28 

33 

83010 

86728 

83821 

86020 

84605 

85287 

27 

34 

83023 

86717 

83834 

86008 

84618 

85274 

26 

a-j 

83037 

86705 

83848 

85996 

84630 

85262 

25 

36 

83051 

86694 

83861 

85984 

84643 

85250 

24 

37 

83065 

86682 

&3874 

85972 

84656 

85237 

23 

38 

83078 

86670 

83887 

85960 

84669 

85225 

22 

39 

83092 

86659 

83901 

85948 

84682 

85212 

21 

40 

9.83106 

9.86647 

9.83914 

9.85936 

9.84694 

9.85200 

20 

41  t 

83120 

866:35 

83927 

85924 

84707 

85187 

19 

42 

83133 

86624 

83940 

8.5912 

84720 

85175 

18 

43 

83147 

86612 

83954 

85900 

84733 

85162 

17 

44 

83161 

86600 

83967 

85888 

84745 

85150 

16 

45 

83174 

86589 

83980 

85876 

84758 

85137 

15 

46 

83i8S 

86577 

83993 

85864 

84771 

85125 

14 

47 

83202 

86565 

84006 

85851 

84784 

85112 

13 

48 

83215 

86554 

84020 

85839 

84796 

85100 

12 

49 

83229 

86542 

84033 

85827 

84809 

85087 

11 

50 

9.83242 

9.86530 

9.84046 

9.a5815 

9.84822 

9.85074 

10 

51 

83256 

86518 

84059 

85803 

84835 

85062 

9 

52 

83270 

86507 

84072 

85791 

84847 

85049 

8 

53 

83283 

86495 

84085 

85779 

84860 

85037 

< 

54 

83297 

86483 

84098 

85766 

84873 

85024 

6 

55 

83310 

86472 

84112 

85754 

84885 

85012 

5 

56 

8:3:324 

86460 

84125 

a5742 

84898 

84999 

4 

57 

8a3.38 

86448 

84138 

857:30 

84911 

84986 

3 

58 

8:33ol 

86436 

84151 

85718 

84923 

84974 

2 

59 

83365 

86425 

84164 

85706 

849:36 

84961 

1 

60 

83.378 

86413 

84177 

85693 

84949 

84949 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

470 

46° 

45° 

Oe*  •-  •  >L  OS* 

Sb 

;.  Sh 

t  0  - 

^, 

TABLE  VIII.— LOG.   TANGENTS    AND   COTANGENTS.    253 


/ 

0" 

1° 

2° 

1   / 

1 

1- 

60 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

—  QO 

00 

8.24192 

11.75808 

8.54308 

11.45692 

1 

6.46373 

13.53627 

24910 

75090 

54669 

45331 

59 

2 

76476 

23524 

25616 

74384 

55027 

44973 

58 

3 

94085 

05915 

26312 

73688 

55382 

44618 

57 

4 

7.06579 

12.93421 

26996 

73004 

55734 

44266 

56 

5 

16270 

83730 

27669 

72331 

56083 

43917 

55 

6 

24188 

75812 

28332 

71668 

56429 

43571 

54 

7 

30882 

69118 

28986 

71014 

66773 

43227 

53 

8 

36682 

63318 

29629 

70371 

57114 

42886 

52 

9 

41797 

58203 

30263 

69737 

57452 

42548 

51 

10 

7.46373 

12.53627 

8.30888 

11.69112 

8.57788 

11.42212 

50 

11 

50512 

49488 

31505 

68495 

58121 

41879 

49 

12 

54291 

45709 

32112 

67888 

58451 

41549 

48 

Vi 

57767 

42233 

32711 

67289 

58779 

41221 

47 

14 

60986 

39014 

33;W2 

66698 

59105 

40895 

46 

15 

63982 

36018 

33886 

66114 

59428 

40572 

45 

16 

66785 

33215 

34461 

65539 

59749 

40251 

44 

17 

69418 

30582 

35029 

64971 

C0068 

39932 

43 

18 

71900 

28100 

35590 

64410 

60384 

39616 

42 

19 

74248 

25752 

36143 

63857 

60698 

39302 

41 

20 

7  76476 

12.23524 

8.36689 

11.63311 

8.61009 

11.38991 

40 

21 

78595 

21405 

37-J29 

62771 

61319 

38681 

39 

22 

80615 

19385 

37762 

62238 

61626 

38374 

38 

23 

82546 

17454 

38289 

61711 

61931 

38069 

37 

24 

84394 

15606 

38809 

61191 

62234 

37766 

36 

25 

86167 

13833 

39323 

60677 

62535 

37465 

35 

26 

87871 

12129 

39832 

60168 

6-2834 

37166 

34 

27 

89510 

10490 

40334 

59666 

63131 

36869 

33 

28 

91089 

08911 

40830 

59170 

63426 

36574 

32 

29 

92C13 

07387 

41321 

58679 

63718 

36282 

31 

30 

7.94086 

12.05914 

8.41807 

11.58193 

8.64009 

11.35991 

30 

31 

95510 

04490 

42287 

57713 

64298 

35702 

29 

32 

96889 

03111 

42762 

57238 

64585 

35415 

28 

33 

98225 

01775 

43232 

56768 

64870 

35130 

27 

34 

99522 

00478 

43696 

56304 

65154 

34846 

26 

35 

8.00781 

11.99219 

44156 

55844 

65435 

34565 

25 

36 

02004 

97996 

44611 

55389 

65715 

34285 

24 

37 

03194 

96806 

45061 

54939 

65993 

34007 

23 

38 

04:353 

95647 

45507 

54493 

66269 

33731 

22 

39 

05481 

94519 

45948 

54052 

66543 

33457 

21 

40 

8.06581 

11.93419 

8.46385 

11.53615 

8.66816 

11.33184 

20 

41 

07653 

92347 

46817 

53183 

67087 

32913 

19 

42 

08700 

91300 

47245 

52755 

67356 

32644 

18 

43 

09722 

90278 

47669 

52331 

67624 

32376  ; 

17 

44 

10720 

89280 

48089 

51911 

67890 

3-2110  ' 

16 

45 

11696 

88304 

48505 

51495 

68154 

31846  1 

15 

46 

12651 

87349 

48917 

51083 

68417 

31583  i 

14 

47 

13585 

86415 

49325 

50675 

68678 

31322 

13 

48 

14500 

85500 

49729 

50271 

68938 

31062 

12 

49 

15395 

84605 

50130 

49870 

69196 

30804 

11 

50 

8.16273 

ll.a3727 

8.50527 

11 .49473 

8.69453 

11.30547 

10 

51 

17133 

82867 

50920 

49080 

69708 

20292 

9 

52 

179:6 

82024 

51310 

4S690 

69962 

30038 

8 

53 

18804 

81196 

51696 

48304 

70214 

29786 

7 

54 

19616 

80384 

52079 

47921 

70465 

29535 

6 

55 

20413 

79587 

52459 

47541 

70714 

29286  1 

5 

56 

21195 

78805 

52835 

47165 

70962 

29038  i 

4 

57 

21964 

78036 

53208 

46792 

71208 

28792  ' 

8 

58 

227:i0 

77280 

53578 

464-22 

714.53 

28547 

2 

59 

23462 

76538 

53945 

46055 

71697 

28303 

1 

60 

24192 

75808 

54308 

45692 

71940 

28060 

0 

/ 

Cotau 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

89° 

88° 

87°      ' 

!o4  TABLE  vni.—LOW.  TANGENTS   AND    COTANGENTS 


/ 

1 

3° 

4» 

5° 

1 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

8.71940 

11.28060 

8.84464 

11.15.536 

8.94195 

11.05805 

60 

1 

72181 

27819 

84646 

15354 

94340 

05660 

59 

2 

72420 

27580 

84826 

1.5174 

94485 

05515 

58 

3 

72659 

27341 

85006 

14994 

94630 

05370 

57 

4 

72896 

27104 

85185 

14815 

94773 

05227 

56 

5 

73182 

26868 

85:363 

14637 

94917 

0.508:? 

55 

6 

73366 

26634 

85540 

14460 

95060 

04940 

54 

7 

73600 

26400 

85717 

14283 

95202 

04798 

53 

8 

73832 

26168 

85893 

14107 

95344 

04656 

52 

9 

74063 

25937 

86069 

13931 

95486 

04514 

51 

10 

8.74292 

11.25708 

8.86243 

11.137.57 

8.9.5627 

11.04373 

50 

i  11 

74521 

25479 

86417 

1.3583 

95767 

04233 

49 

1  12 

74748 

25252 

86591 

13409 

95908 

04092 

48 

13 

74974 

25026 

86763 

132.37 

96047 

03953 

47 

14 

75199 

24801 

86935 

13065 

96187 

03813 

46 

15 

75423 

24577 

87106 

12894 

96325 

03675 

45 

16 

75645 

24355 

87277 

12723 

96464 

03536 

44 

17 

75867 

24133 

87447 

12553 

96C02 

03398 

43 

18 

76087 

23913 

87616 

12.384 

96739 

03261 

42 

19 

76306 

23694 

8r;'85 
8.87953 

12215 
11.12047 

96877 
8.97013 

41 
40 

20 

3.76525 

11.23475 

11.02987 

21 

76742 

23258 

88120 

11880 

97150 

02850 

39 

22 

76958 

23042 

88287 

11713 

97285 

02715 

38 

23 

77173 

22827 

88453 

11547 

97421 

02579 

37 

24 

77387 

22613 

88618 

11382 

97556 

02444 

36 

25 

77600 

22400 

88783 

11217 

97691 

02309 

35 

26 

77811 

22189 

88948 

11052 

97825 

02175 

34 

27 

78022 

21978 

89111 

10SS9 

979.59 

02041 

33 

28 

78232 

21768 

89274 

10726 

98092 

01908 

32 

29 

78441 

21559 

89437 

10563 

98225 

01775 

31 

30 

8.78649 

11.21351 

8.89598 

11.10402 

8.98.3.58 

11.01642 

30 

31 

78855 

21145 

89760 

10240 

98490 

01510 

29 

32 

79061 

20939 

89920 

10080 

98622 

01378 

28 

33 

79266 

20734 

90080 

09920 

987.53 

01247 

27 

34 

79470 

20530 

90240 

09760 

98884 

01116 

26 

35 

79673 

20827 

90399 

09601 

99015 

00985 

25 

36 

79S75 

201S5 

90557 

09443 

99145 

00855 

24 

37 

80076 

19924 

90715 

09285 

99275 

00725 

23 

38 

80277 

19723 

90872 

09128 

99405 

00595 

22 

39 

80476 

•  19524 

91029 

08971 

99534 

00466 

21 

40 

8.80674 

11.19326 

8.91185 

11.08815 

8.99662 

11.00.338 

20 

41 

80872 

19128 

91340 

08660 

99791 

00209 

19 

42 

81068 

18932 

91495 

08505 

99919 

00081 

18 

43 

81264 

18736 

916.50 

0a350 

9.00046 

10.99954 

17 

44 

81459 

18541 

91803 

08197 

00174 

99826 

16 

45 

816.53 

18317 

91957 

08043 

00301 

99699 

15 

46 

81H46 

18154 

92110 

07890 

00427 

99573 

14 

1  47 

82038 

17962 

92262 

0773S 

00553 

99447 

IS 

48 

82230 

17770 

92414 

07586 

00679 

99321 

12 

49 

82420 

17580 

92565 

07435 

00805 

99195 

11 

50 

8.82610 

11.17390 

8.92716 

11.07284 

9.00930 

10.99070 

10 

51 

82799 

17201 

92866 

07134 

010.i5 

98945 

9 

52 

82987 

17013 

93016 

06984 

01179 

98821 

8 

53 

83175 

16825 

93165 

06835 

01303 

98697 

51 

83361 

166.39 

93313 

06587 

01427 

98573 

6 

5.-. 

83547 

164.53 

93462 

06538 

015.50 

98450 

5 

56 

83732 

16268 

93609 

06391 

01673 

98327 

4 

57 

83916 

16084 

93756 

06244 

01796 

98204 

3 

58 

84100 

1.590O 

93903 

06097 

01918 

9^^082 

2 

59 

84282 

15718 

94049 

05951 

02040 

97960  ^ 

1 

60 

84464 

15536 

94195 

05805 

02162 

978.38 

0 

/ 

Co  tan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

S6» 

8o° 

84° 

TABLE  VIII.— LOG.  TANGENTS   AND  COTANGENTS.   255 


6 

B 

7 

o 

8° 

1 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 
10.85220 

0 

9.02162 

10.97838 

9.08914 

10.91086 

9.14780 

60 

1 

02283 

97717 

09019 

90981 

14872 

85128 

59 

o 

02404 

97596 

09123 

90877 

14963 

85037 

58 

3 

02525 

97475 

09227 

90773 

15054 

84946 

57 

4 

02645 

97355 

09330 

90670 

15145 

84855 

56 

5 

02766 

97234 

09434 

90566 

15236 

84764 

55 

6 

02885 

97115 

09537 

90463 

15327 

84673 

54 

i 

03005 

96995 

09640 

90360 

15417 

84583 

53 

8 

03124 

96876 

09742 

90258 

15508 

84492 

52 

9 

03242 

96758 

09845 

90155 

15598 

84402 

51 

10 

9.03361 

10.96639 

9.09947 

10.90053 

9.15688 

10.84312 

50- 

11 

03479 

96521 

10049 

89951 

15777 

84228 

49 

12 

03597 

96403 

10150 

89850 

15867 

84133 

48 

13 

03714 

96286 

10252 

89748 

15956 

84044 

47 

14 

03832 

96168 

10353 

89647 

16046 

83954 

46 

15 

03948 

96052 

10454 

89546 

16135 

83865 

45 

16 

04065 

95935 

10555 

89445 

16224 

83776 

44 

17 

04181 

95819 

10656 

89344 

16312 

83688 

43 

18 

04297 

95703 

10756 

89244 

16401 

83599 

42 

19 

04413 

95587 

10856 

89144 

16489 

83511 

41 

20 

9.04528 

10.95472 

9.10956 

10.89044 

9.16577 

10.83423 

40 

21 

04643 

95357 

11056 

88944 

16665 

83335 

39 

22 

04758 

95242 

11155 

88845 

16753 

83247 

38 

23 

04873 

95127 

11254 

88746 

16841 

83159 

37 

24 

04987 

95013 

11353 

88647 

16928 

83072 

36 

25 

05101 

94899 

11452 

88548 

17016 

82984 

35 

26 

05214 

94786 

11551 

88449 

17103 

82897 

34 

27 

05328 

94672 

11649 

88351 

17190 

82810 

33 

28 

05441 

94559 

11747 

88253 

17277 

82723 

32 

29 

05553 

94447 

11845 

88155 

17363 

82637 

31 

30 

9.05666 

10.94334 

9.11943 

10.88057 

9.17450 

10.82550 

30 

31 

05778 

94222 

12040 

87960 

17536 

82464 

29 

32 

05890 

94110 

12138 

87862 

17622 

82378 

28 

33 

06002 

93998 

12235 

87765 

17708 

82292 

27 

34 

06113 

93887 

12332 

87668 

17794 

82206 

26 

35 

06224 

98776 

12428 

87572 

17880 

82120 

25 

36 

06335 

93665 

12525 

87475 

17965 

82035 

24 

37 

06445 

93555 

12621 

87379 

18051 

81949 

23 

38 

06556 

93444 

12717 

87283 

18136 

81864 

22 

39 

06666 

93334 

12813 

87187 

18221 • 

81779 

21 

40 

9.06775 

10.93225 

9.12909 

10.87091 

9.18306 

10.81694 

20 

41 

06885 

93115 

13004 

86996 

18391 

81609 

19 

42 

06994 

93006 

13099 

86901 

18475 

81525 

18 

43 

07103 

92897 

13194 

86806 

18560 

81440 

17 

44 

07211 

92789 

13289 

86711 

18644 

81356 

16 

45 

07320 

92680 

13384 

806 16 

18728 

81272 

15 

46 

07428 

92572 

13478 

86522 

18812 

81188 

14 

47 

07536 

92464 

13573 

86427 

18896 

81104 

13 

48 

07643 

92357 

13667 

86338 

18979 

81021 

12 

49 

07751 

92249 

13761 

86239 

19063 

80937 

11 

50 

9.07858 

10.92142 

9.13854 

10.86146 

9.19146 

10.80854 

10 

51 

07964 

92036 

13948 

86052 

19229 

80771 

9 

52 

08071 

91920 

14041 

85959 

19312 

80688 

8 

53 

08177 

91823 

14134 

85866 

19395 

80605 

^ 

54 

08283 

91717 

14227 

85773 

19478 

80522 

6 

55 

08389 

91(n) 

14320 

85680 

19561 

80439 

5 

56 

08495 

91505 

14412 

85538 

19643 

80357 

4 

57 

08600 

91400 

14504 

85496 

19725 

80275 

3 

58 

08705 

91295 

1 4597 

85403 

19807 

80193 

2 

59 

08810 

91190 

14688 

85312 

19889 

80111 

1 

60 

08914 

91086 

14780 

85220 

19971 

80029 

0 

1 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

83" 

82" 

81° 

256  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

9° 

10" 

11° 

t 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.19971 

10.80029 

9.24632 

10.75368 

9.28865 

10.71135 

60 

1 

20053 

79947 

24706 

75294 

289:33 

71067 

59 

2 

20134 

79866 

24779 

75221 

29000 

71000 

58 

3 

20216 

79784 

24853 

75147 

29067 

70933 

57 

1 

20297 

79703 

24926 

75074 

29134 

70866 

56 

5 

20378 

79622 

25000 

75000 

29201 

70799 

55 

6 

20459 

79541 

25073 

74927 

29268 

70732 

54 

7 

20540 

79460 

25146 

74854 

29335 

70665 

53 

8 

20621 

79379 

25219 

74781 

29402 

70598 

52 

9 

20701 

79299 

25292 

74708 

29468 

70532 

51 

10 

9.20782 

10.79218 

9.25365 

10.74685 

9.29535 

10.70465 

50 

11 

20862 

79138 

25437 

74563 

29601 

70399 

49 

12 

20942 

79058 

25510 

74490 

29668 

70332 

48 

13 

21022 

78978 

25582 

74418 

29734 

70266 

47 

14 

21102 

78898 

25655 

74345 

29800 

70200 

46 

15 

21182 

78818 

25727 

74273 

29866 

70134 

45 

16 

21261 

78739 

25799 

74201 

29932 

70068 

44 

17 

21341 

78659 

25871 

74129 

29998 

70002 

43 

18 

21420 

78580 

25943 

74057 

30064 

69936 

42 

19 

21499 

78501 

26015 

73985 

30130 

69870 

41 

30 

9.21578 

10.78422 

9.26086 

10.73914 

9.30195 

10.69805 

40 

21 

21657 

78343 

26158 

73842 

30261 

69739 

39 

22 

21736 

78264 

26229 

73771 

30326 

696T4 

38 

23 

21814 

78186 

26301 

73699 

30391 

69609 

37 

24 

21893 

78107 

26372 

73628 

30457 

69543 

36 

25 

21971 

78029 

26443 

7:3657 

30522 

69478 

35 

26 

22049 

77951 

26514 

73486 

:30587 

69413 

34 

27 

22127 

77873 

26585 

73415 

30652 

69348 

33 

28 

22205 

77795 

266.55 

7:3345 

30717 

69283 

32 

29 

82283 

77717 

26726 

73274 

30782 

69218 

31 

30 

9.22361 

10.77639 

9.26797 

10.73203 

9.30S46 

10.691.54 

30 

31 

22438 

77562 

26^67 

731:33 

30911 

69089 

29 

32 

22516 

77484 

26937 

73063 

30975 

69025 

28 

33 

22593 

77407 

27008 

72992 

31040 

68960 

27 

34 

22670 

773;J0 

27078 

72922 

31104 

68896 

26 

35 

22747 

77253 

27148 

72852 

31168 

68832 

25 

36 

22824 

77176 

27218 

72782 

31233 

68767 

24 

37 

22901 

77099 

27288 

72712 

31297 

68703 

23 

38 

22977 

77023 

27357 

72643 

31361 

68639 

22 

39 

2.i034 

76946 

27427 

72573 

31425 

68575 

21 

40 

9.23130 

10.76870 

9.27496 

10.72.504 

9.31489 

10.68511 

20 

41 

23206 

76794 

27566 

72434 

315.52 

68448 

19 

42 

23283 

76717 

27635 

72365 

31616 

68384 

18 

43 

23359 

76641 

27704 

72296 

31679 

68321 

17 

44 

^3435 

76565 

27773 

72227 

31743 

68257 

16 

45 

23510 

76490 

27842 

72158 

31806 

68194 

15 

46 

23586 

76414 

2791 1 

72089 

31870 

681:30 

14 

47 

2:3661 

76;i39 

27980 

72020 

319:33 

68067 

13 

48 

23737 

76263 

28049 

71951 

31996 

68004 

12 

49 

23812 

76188 

28117 

71883 

32059 

67941 

11 

50 

9.2:J887 

10.76113 

9.2S1S6 

10.71S14 

9.32122 

10.67878 

10 

51 

23962 

76038 

28254 

71746 

:32185 

67815 

9 

52 

24037 

75963 

28323 

71677 

:32248 

67752 

8 

m 

24112 

75888 

28391 

71609 

3-'3l  1 

676.e9 

^ 

i 

54 

24186 

7.5814 

28459 

71541 

:32373 

67627 

6 

55 

24261 

75739 

28527 

71473 

324:30 

67.564 

5 

.56 

243:i-> 

7.5665 

2859.5 

71405 

32498 

67502 

4 

57 

24410 

75590 

28662 

71338 

3^561 

K7439 

3 

58  ! 

24484 

7.5516 

28730 

71270 

32623 

67377 

o 

59 

21.5.58 

7.5442 

28708 

71202 

32685 

67315 

1 

60  1 

\ 

24632 

75368 

28865 

71135 

32747 

672.53 

0 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

so- 

JJP 

78° 

TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS.   257 


/ 

12° 

13° 

14° 

1 
/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.32747 

10.67253 

9.36336 

10.63664 

9.39677 

10.60323 

60 

1 

32810 

67190 

36394 

63606 

39731 

60269 

59 

2 

32872 

67128 

36452 

63548 

39785 

60215 

58 

3 

32933 

670G7 

36509 

63491 

39838 

60162 

.57 

4 

32995 

67005 

36566 

63434 

89892 

60108 

56 

5 

33057 

66943 

36624 

63376 

39945 

60055 

55 

6 

33119 

6G881 

36681 

63319 

39999 

60001 

54 

r^ 
^ 

83180 

66820 

36738 

63262 

40052 

59948 

53 

8 

33-242 

66758 

36795 

63205 

40106 

59894 

52 

9 

33303 

66697 

36852 

63148 

40159 

59841 

51 

10 

9.33365 

10.66C35 

9.36909 

10.63091 

9.40212 

10.59788 

50 

11 

33426 

66574 

3G9G6 

63034 

40266 

59734 

49 

12 

33487 

66513 

37023 

62977 

40319 

59681 

48 

13 

33548 

66452 

37080 

62920 

40372 

59628 

47 

14 

33G09 

66391 

37137 

62863 

40425 

59575 

46 

15 

33670 

66330 

37193 

62807 

40478 

59522 

45 

16 

33731 

66269 

37250 

62750 

40531 

59469 

44 

17 

33792 

66208 

37306 

62694 

40584 

59416 

43 

18 

33853 

6G147 

37363 

62637 

40636 

59364 

42 

19 

33913 

66087 

37419 

62581 

40689 

59311 

41 

20 

9.33974 

10.66026 

9.37476 

10.62524 

9,40742 

10.59258 

40 

21 

34034 

65966 

37532 

62468. 

40795 

59205 

39 

22 

34095 

65905 

37588 

62412 

40847 

591,53 

38 

23 

34155 

65845 

37644 

62356 

40900 

59100 

37 

24 

34215 

65785 

37700 

62300 

40952 

59048 

36 

25 

34276 

65724 

37756 

62244 

41005 

58995 

35 

26 

34336 

65664 

37812 

62188 

41057 

58943 

34 

27 

34396 

65604 

37868 

621.32 

41109 

58891 

33 

28 

34456 

65544 

37924 

62076 

41161 

58839 

32 

29 

34516 

65484 

37980 

62020 

41214 

58786 

31 

30 

9.34576 

10.65424 

9.38035 

10.61965 

9.41266 

10.. 587.34 

30 

31 

34635 

65365 

38091 

61909 

41318 

58682 

29 

;32 

34695 

65305 

38147 

61853 

41370 

58630 

28 

33 

34755 

6.5245 

38202 

61798 

41422 

58578 

27 

34 

34814 

65186 

38257 

61743 

41474 

58526 

26 

35 

34874 

65126 

38313 

61687 

41526 

58474 

25 

36 

34933 

65067 

38368 

61632 

41578 

58422 

24 

37 

34992 

65008 

3&423 

61577 

41629 

58371 

23 

38 

35051 

64949 

38479 

61521 

41681 

58319 

22 

39 

35111 

64889 

38534 

61466 

41733 

58267 

21 

40 

9.35170 

10.6-4830 

9.38589 

10.61411 

9.41784 

10.58216 

20 

41 

35229 

64771 

38644 

61356 

41836 

.58164 

19 

42 

35288 

64712 

.38699 

61301 

41887 

58113 

18 

43 

35347 

64G53 

3-^754 

61246 

41939 

58061 

17 

44 

35405 

64595 

38808 

61192 

41990 

58010 

16 

45 

35464 

64536 

38863 

61137 

42041 

57959 

15 

46 

35523 

64477 

38918 

61082 

42093 

57907 

14 

47 

35581 

64419 

3S972 

61028 

42144 

57856 

13 

48 

35640 

64360 

39027 

60973 

42195 

57805 

12 

49 

35G98 

64302 

39082 

60918 

42246 

57754 

11 

50 

9.35757 

10.64243 

9.39136 

10.60864 

9.42297 

10.. 57703 

10 

51 

35815 

64185 

39190 

60810 

42348 

57652 

9 

52 

35873 

64127 

.39245 

60755 

4239!» 

57601 

8 

53 

35931 

64069 

39299 

60701 

424.50 

57550 

7 

54 

35989 

64011 

39.^53 

60647 

42501 

57499 

6 

55 

36047 

63953 

.30407 

60593 

425,52 

57448 

5 

56 

36105 

63895 

39461 

60539 

42603 

57397 

4 

57 

36163 

63837 

39515 

60485 

42653 

57.347 

3 

58 

36221 

63779 

39569 

60431 

42704 

57296 

2 

59 

36279 

63721 

39623 

60377 

427.55 

57245 

1 

60 

36336 

63664 

39677 

60323 

42805 

57195 

0 

1   • 

/ 

Cotau 

Tan 

Cotan 

Tan 

Cotan 

Tan 

77° 

76° 

75° 

358    TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


^ 

15» 

16° 

17° 

1 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.42805 

10.57195 

9.45750 

10.54250 

9.48534 

10.51466 

60 

1 

42856 

57144 

45797 

54203 

48579 

51421 

59 

2 

42906 

57094 

45845 

54155 

48624 

51376 

58 

3 

42957 

57043 

45892 

54108 

48669 

51331 

57 

4 

43007 

56993 

45940 

54060 

48714 

51286 

56 

5 

43057 

56943 

45987 

54013 

48759 

51241 

55 

6 

43108 

56892 

46035 

53965 

48804 

51196 

54 

7 

43158 

56842 

46082 

53918 

48849 

51151 

53 

8 

43208 

56792 

46130 

53870 

48894 

51106 

52 

9 

43258 

56742 

46177 

53823 

48939 

51061 

51 

10 

9.43308 

10.56692 

9.46224 

10.53776 

9.48984 

10.51016 

50 

11 

43358 

56642 

46271 

53729 

49029 

50971 

49 

12 

43408 

56592 

46319 

53681 

49073 

50927 

48 

13 

43458 

56542 

46366 

53634 

49118 

50882 

47 

14 

43508 

56492 

46413 

53.587 

49163 

50837 

46 

15 

48558 

56442 

46460 

53540 

49207 

50793 

45 

16 

43607 

56393 

46507 

53493 

49252 

50748 

44 

17 

43657 

56343 

46554 

53446 

49296 

50704 

43 

18 

43707 

56293 

46601 

53399 

49341 

50659 

42 

19 

43756 

56244 

46648 

53352 

49385 

50615 

41 

20 

9.43806 

10.56194 

9.46694 

10  53306 

9.49430 

10.50570 

40 

21 

43855 

56145 

46741 

53259 

49474 

50526  ■ 

39 

22 

43905 

56095 

46788 

53212 

49519 

50481 

38 

23 

43954 

56046 

46835 

53165 

4956:3 

50437 

37 

24 

44004 

55996 

46881 

53119 

49607 

50393 

36 

25 

44053 

55947 

46928 

53072 

49652 

50348 

35 

26 

44102 

55898 

46975 

53025 

49(i96 

50304 

34 

27 

44151 

55849 

47021 

52979 

49740 

50260 

33 

28 

44201 

55799 

47068 

52932 

49784 

50216 

32 

29 

44250 

55750 

47114 

52886 

49828 

50172 

31 

30 

9.44299 

10.55701 

9.47160 

10.52840 

9.49872 

10.50128 

30 

31 

44348 

55652 

47207 

527J3 

49916 

50084 

29 

32 

443!)7 

55603 

47253 

52747 

49960 

50040 

28 

83 

44446 

55554 

47299 

52701 

50004 

49996 

27 

34 

44495 

55505 

47346 

526*4 

50048 

49952 

26 

35 

44544 

55456 

47392 

52608 

50092 

49908 

25 

36 

44592 

55408 

47438 

52562 

50136 

49864 

24 

37 

44641 

55359 

474S4 

52516 

.50180 

49820 

23 

38 

44690 

55310 

47530 

52470 

.5(IJ23 

49777 

22 

39 

44738 

55262 

47576 

52424 

50267 

49733 

21 

40 

9.447H7 

10.55213 

9.47622 

10.52378 

9.50311 

10.49689 

20 

41 

44836 

55164 

47668 

52332 

50355 

49645 

19 

42 

44884 

55116 

47714 

52286 

50398 

49602 

18 

43 

44933 

.55067 

47760 

52240 

50442 

49558 

17 

44 

44981 

55019 

47806 

52194 

50485 

49515 

16 

45 

4502y 

54971 

47852 

52148 

.50529 

49471 

15 

46 

45078 

54922 

47897 

52103 

50572 

49428 

14 

47 

451 2(; 

54874 

47943 

52057 

.50616 

49384 

13 

48 

45174 

54826 

479S9 

52011 

50659 

49341 

12 

49 

45222 

54778 

48035 

51965 

50703 

49297 

11 

50 

9.45271 

10.54729 

9.48080 

10.51920 

9.50746 

-  10.49254 

10 

51 

45319 

546S1 

4Sl-,'6 

51874 

50789 

49211 

9 

52 

45367 

54633 

48171 

51829 

50833 

491 07 

8 

53 

45415 

54585 

48217 

51783 

50876 

49124 

7 

51 

45463 

54537 

48262 

51738 

50919 

49081 

6 

55 

45511 

54489 

48307 

5 '693 

50962 

49038 

5 

56 

45559 

54441 

483.53 

51647 

51005 

48995 

4 

57 

45()06 

54394 

48-598 

51602 

51048 

4S952 

3 

58 

45654 

54346 

48443 

51557 

51092 

48908 

2 

59 

-  45702 

54298 

48489 

51511 

51135 

48*^65 

1 

60 

45750 

54250 

48534 

51466 

51178 

48822 

0 

/ 

Co  I  an 

Tail 

Cotan 

Tan 

Cotan 

Tan 

/ 

740 

73° 

72° 

TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS.   259 


/ 

18» 

19° 

20° 

/ 

Tan 

Cotaii 

Tan 

Cotan 

Tan 

Cotan 

0 

9.51178 

10.48822 

9.53697 

10.46303 

9.56107 

10.4:3893 

60 

1 

51221 

48779 

53738 

46263 

56146 

43854 

59 

o 

51264 

48736 

53779 

46221 

56185 

43815 

58 

i 

51306 

48694 

53820 

46180 

56224 

43776 

57 

4 

51349 

48651 

53861 

46139 

56264 

43736 

56 

5 

51392 

48608 

53902 

46098 

56303 

43697 

55 

6 

51435 

48565 

53943 

46057 

56342 

43658 

54 

7 

51478 

48522 

53984 

46016 

56381 

43619 

53 

8 

51520 

48480 

54025 

45975 

56420 

43580 

52 

9 

51563 

48437 

54065 

45935 

56459 

43541 

51 

10 

9.51606 

10.48394 

9.54106 

10.4.5894 

9.56498 

10.43502 

50 

11 

51648 

48352 

54147 

45853 

56,537 

43463 

49 

12 

51691 

48309 

54187 

45813 

56576 

43424 

48 

13 

51734 

48266 

54228 

45772 

56815 

43385 

47 

14 

51776 

48224 

54269 

45731 

56654 

43346 

46 

15 

51819 

48181 

54309 

45691 

56693 

43307 

45 

16 

51861 

48139 

54350 

45650 

56738 

43268 

44 

17 

51903 

48097 

54390 

45610 

56771 

43229 

43 

18 

51946 

48054 

54431 

45569 

56810 

43190 

42 

19 

51988 

48012 

54471 

45529 

56849 

43151 

41 

20 

9.52031 

10.47969 

9.54512 

10.45488 

9.56887 

10.43113 

40 

21 

52073 

47927 

545.52 

45448 

56926 

43074 

39 

22 

52115 

47885 

54593 

45407 

56965 

43035 

38 

28 

52157 

47843 

54033 

45367 

57004 

42996 

37 

24 

52200 

47800 

54673 

45327 

57042 

42958 

86 

25 

52242 

47758 

54714 

45286 

57081 

42919 

35 

26 

52284 

47716 

54754 

45246 

57120 

42880 

34 

27 

52326 

47674 

54794 

45206 

57158 

42842 

33 

28 

52368 

47032 

54835 

45165 

57197 

42803 

32 

29 

52410 

4<590 

54875 

45125 

57235 

42765 

31 

30 

9.52452 

10.47548 

9.54915 

10.45085 

9.57274 

10.42726 

30 

31 

52494 

47506 

54955 

45045 

5^312 

42688 

29 

32 

52536 

47464 

54995 

45005 

57351 

42649 

28 

33 

52578 

47422 

55035 

44965 

57389 

42611 

27 

34 

52620 

4^380 

55075 

44925 

57428 

42572 

26 

35 

52661 

47339 

5.5115 

44885 

57466 

42534 

25 

36 

52703 

47J97 

551.55 

44845 

57.504 

42496 

24 

37 

52745 

47255 

55195 

44805 

57543 

42457 

23 

38 

52787 

47213 

55235 

44765 

57581 

42419 

22 

39 

52829 

47171 

5.5275 

44725 

57619 

42381 

21 

40 

9.. 52870 

10.47130 

9.5.5315 

10.44685 

9.57658 

10.42342 

20 

41 

52912 

47088 

5.^3:)5 

44645 

57696 

42304 

19 

42 

52953 

47047 

5.5395 

44605 

57734 

42266 

18 

43 

52905 

47005 

55434 

44566 

57772 

42228 

17 

44 

53037 

46963 

55474 

445-.'6 

57810 

42190 

16 

45 

53078 

4<;922 

5.5514 

44486 

57849 

421.51 

15 

46 

53120 

46880 

5.5.554 

44446 

57887 

42113 

14 

47 

53161 

46839 

55593 

44407 

57925 

42075 

13 

48 

53202 

46798 

55633 

44367 

57963 

42037 

12 

49 

53244 

467.56 

55673 

44327 

58001 

41999 

11 

50 

9.53285 

10.46715 

9.5.5712 

10.44288 

9.580.39 

10.41961 

10 

51 

53327 

46673 

55752 

44248 

58077 

41923 

9 

52 

533()8 

46632 

55791 

44209 

.58115 

41885 

8 

53 

53409 

46.591 

55S31 

44169 

581.53 

41847 

54 

53450 

46.550 

55870 

441.30 

.58191 

41809 

6 

55 

53492 

46.508 

5.5910 

44090 

58229 

41771 

5 

56 

53533 

46467 

.55949 

44051 

582(17 

41:33 

4 

57 

53574 

46426 

55989 

44011 

58304 

41696 

3 

58 

53615 

46385 

56028 

43972 

.58342 

41658 

2 

59 

53656 

46.344 

56067 

43933 

58380 

41620 

1 

60 

63697 

46303 

56107 

43893 

58418 

41582 

0 

/ 

Cotan 

Tan 

Cotan 

Tail 

Cotan 

Tan 
69° 

/ 

71° 

70° 

"60  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

i 

21° 

22° 

23° 

/ 

7 
i 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.58418 

10.41582 

9.60641 

10.39359 

9.62785 

10.37215 

60 

1 

58155 

41545 

60677 

39323 

62820 

37180 

59 

2 

5S493 

41507 

60714 

39286 

62855 

37145 

58 

3 

58531 

41469 

60750 

39250 

62890 

37110 

57 

4 

5S569 

41431 

60786 

39214 

62926 

37074 

56 

5 

58606 

41394 

60823 

39177 

62961 

37039 

55 

6 

58644 

41356 

60859 

39141 

62996 

37004 

54 

7 

58681 

41319 

60895 

.39105 

63031 

36909 

53 

8 

58719 

41281 

60931 

39069 

6.3066 

36934 

52 

9 

58757 

41243 

60967 

39033 

63101 

36899 

51 

10 

9.58794 

10.41206 

9.61004 

10.38996 

9.631.35 

10.36865 

50 

11 

58832 

41168 

61040 

38960 

63170 

368,30 

49 

12 

58869 

41131 

61076 

38924 

63205 

.36795 

48 

13 

58907 

41093 

61112 

38888 

63240 

36760 

47 

14 

58944 

41056 

61148 

38852 

63275 

36725 

46 

15 

58981 

41019 

61184 

38816 

6.3310 

36690 

45 

16 

59019 

40981 

61220 

.38780 

63345 

36655 

44 

17 

59056 

40i)44 

61256 

38744 

63379 

36621 

43 

18 

59094 

40906 

61292 

38708 

63414 

36586 

42 

19 

59131 

40869 

61328 

38672 

63449 

36551 

41 

20 

9.59168 

10.40832 

9.61364 

10.38636 

9.63484 

10. .36516 

40 

21 

59205 

40795 

61400 

38600 

63519 

.36481 

39 

22 

59243 

40757 

614.36 

38564 

63553 

36447 

38 

23 

59280 

40720 

61472 

38528 

63588 

36412 

37 

24 

59317 

40683 

61508 

3f'492 

63623 

.36377 

36 

25 

59354 

40646 

61544 

384.56 

63657 

36313 

35 

26 

59391 

40609 

61.579 

.38421 

63692 

36.308 

34 

27 

59429 

40571 

61615 

38385 

63726 

36274 

.33 

28 

59466 

40534 

616.51 

.38.549 

63761 

.36239 

32 

29 

59503 

40497 

61687 

.38313 

63796 

36204 

31 

30 

9.59540 

10.40460 

9.61722 

10.38278 

9.6.38.30 

10.36170 

30 

31 

59577 

40423 

61758 

3S242 

6.3865 

36135 

29 

32 

59614 

40386 

61794 

38206 

6.3899 

36101 

28 

33 

59651 

40349 

61830 

38170 

63934 

36066 

27 

34 

59688 

40312 

61S65 

3S1.35 

63968 

36032 

26 

35 

59725 

40275 

61901 

38099 

64003 

35997 

25 

36 

59702 

40238 

61936 

38064 

64037 

35963 

24 

37 

59799 

40201 

01972 

38028 

64072 

35928 

23 

38 

59S35 

40165 

62008 

37992 

64106 

35894 

22 

39 

598';2 

40128 

02043 

379.57 

64140 

35860 

21 

40 

9.. 59909 

10.40091 

9.62079 

10.37921 

9.64175 

10.35825 

20 

41 

.59946 

40054 

62114 

37886 

64209 

35791 

19 

42 

59983 

40017 

62150 

37850 

64243 

35757 

18 

43 

60019 

39981 

62185 

37815 

64278 

35722 

17 

44 

60056 

39944 

62221 

37779 

64312 

35688 

16 

45 

60093 

39907 

6225(5 

37744 

64346 

35654 

15 

1 

46 

60130 

39S70 

62292 

37708 

64381 

.3.5619 

14 

47 

60166 

39834 

62327 

37673 

64415 

35585 

13 

48 

60203 

39797 

62302 

.37638 

64449 

35551 

12 

49 

60:i40 

39760 

6239H 

37602 

64483 

35517 

11 

50 

9.00276 

10.39724 

9.62433 

10.. 37.567 

9.64.517 

10.. 35483 

10 

51 

60313 

39687 

62468 

37.5.32 

64.552 

.3.5448 

9 

52 

6034'.» 

39651 

62504 

37496 

64586 

3.5414 

8 

53 

60386 

39614 

62539 

37461 

64620 

3.5380 

i 

54 

00  4. '2 

39.=.78 

62574 

37426 

64654 

35346 

6 

55 

60459 

39.541 

62609 

37391 

64688 

.3.5312 

5 

56 

60405 

39505 

62645 

37.3.55 

64722 

35278 

4 

57 

60532 

39468 

62680 

37320 

64756 

35244 

3 

58 

60568 

394.32 

62715 

37285 

64790 

35210 

') 

59 

60605 

39395 

62750 

372.50  ■ 

64824 

35176 

1  1 

60 

60611 

39359 

62785 

37215 

64858 

35142 

0  1 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

68° 

67° 

66° 

TABLE  VIII. 

LOG. 

TANGENTS  AND 

COTANGENTS 

.  261 

/ 

24° 

1 

26° 

26° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.64858 

10.35142 

9.66867 

10.33133 

9.68818 

10.31182 

60 

1 

64892 

35108 

66900 

33100 

68850 

31150 

59 

64926 

35074 

66933 

33067 

68882 

31118 

58 

3 

649()0 

35040 

66966 

33034 

68914 

31086 

57 

4 

64994 

35006 

66999 

33001 

68946 

31054 

56 

5 

65028 

34972 

67032 

32968 

68978 

31022 

55 

6 

65062 

34938 

67065 

32935 

69010 

30990 

54 

7 

65096 

34904 

67098 

32902 

69042 

30958 

53 

8 

65130 

31870 

67131 

32869 

69074 

30926 

52 

9 

65164 

34836 

67163 

32837 

69106 

30894 

61 

10 

9.65197 

10.34803 

9.67196 

10.32804 

9.69138 

10.30862 

50 

11 

65'i31 

34769 

67229 

32771 

69170 

30830 

49 

12 

65265 

34735 

67262 

32738 

69202 

80798 

48 

13 

65299 

34701 

67295 

32705 

69234 

30766 

47 

14 

65333 

34667 

67327 

32673 

69266 

30734 

46 

15 

65366 

34634 

67360 

3^640 

69298 

30702 

45 

16 

65400 

34600 

67393 

32607 

69329 

30671 

44 

17 

65434 

34566 

67426 

32574 

69361 

30639 

43 

18 

65467 

34533 

67458 

32542 

69393 

30607 

42 

.  19 

65501 

34499 

67491 

32509 

69425 

30575 

41 

20 

9.65535 

10.34465 

9.67524 

10.32476 

9.69457 

10.30543 

40 

21 

65568 

34432 

67556 

32444 

69488 

30512 

39 

22 

65602 

34398 

67589 

32411 

69520 

30480 

38 

23 

65636 

34364 

67622 

32378 

69552 

30448 

37 

24 

65669 

34331 

67654 

32346 

69584 

30416 

36 

25 

65703 

34297 

67687 

32313 

69615 

30385 

35 

26 

65736 

34264 

67719 

32281 

69647 

30353 

34 

27 

65770 

34230 

67752 

32248 

•69679 

30321 

33 

28 

65803 

34197 

67785 

32215 

69710 

30290 

32 

29 

65837 

34163 

67817 

32183 

69742 

30258 

31 

30 

9.65870 

10.34130 

9.67850 

10.32150 

9.69774 

10.30226 

30 

31 

65904 

34096 

67882 

32118 

69805 

30195 

29 

32 

65937 

34063 

67915 

32085 

69837 

30163 

28 

33 

65971 

34029 

67947 

32053 

69868 

30132 

27 

34 

66004 

33996 

67980 

32020 

69900 

30100 

26 

35 

66038 

33962 

68012 

31988 

69932 

30068 

25 

36 

66071 

33929 

08044 

31956 

69963 

30037 

24 

37 

66104 

33896 

68077 

31923 

69995 

30005 

23 

38 

66138 

33862 

68109 

31891 

70026 

29974 

22 

39 

66171 

33829 

68142 

31858 

70058 

29942 

21 

40 

9.66204 

10.33796 

9.68174 

10.31826 

9.70089 

10.29911 

20 

41 

66238 

33762 

68206 

31794 

70121 

29879 

19 

42 

66271 

33729 

68239 

31761 

70152 

29848 

18 

43 

66304 

33696 

68271 

31729 

70184 

29816 

17 

44 

66337 

33663 

68303 

31697 

70215 

29785 

16 

45 

66371 

33629 

68336 

31664 

70247 

29753 

15 

46 

66104 

33596 

68368 

31632 

70278 

29722 

14 

47 

66437 

33563 

68400 

31600 

70309 

29691 

13 

48 

66470 

33530 

68432 

31568 

70341 

29659 

12 

49 

66503 

33497 

68465 

31535 

70372 

29628 

11 

50 

9.66537 

10.33463 

9.68497 

10.31503 

9.70404 

10.29596 

10 

51 

66570 

33430 

68529 

31471 

70435 

29565 

9 

52 

66603 

33397 

68561 

31439 

70466 

29534 

8 

53 

66636 

33364 

68593 

31407 

70498 

29502 

7 

54 

60669 

33331 

68626 

31374 

70529 

29471 

6 

55 

66702 

33298 

68658 

31342 

70560 

29440 

5 

56 

66735 

33265 

68690 

31310 

70592 

29408 

4 

57 

66768 

33232 

68722 

31278 

70623 

29377 

8 

58 

66801 

33199 

68754 

31246 

70654 

29346 

2 

59 

66834 

33166 

68786 

31214 

70685 

29315 

1 

60 

/ 



66867 

33133 

68818 

31182 

70717 

29283 

0 

Cntan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

66» 

64° 

68° 

262  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


t 

2 

7° 

* 
* 

>8° 

2 

9° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.70717 

10.29283 

9.72567 

10.27433 

9.74375 

10.25625 

60 

1 

70748 

29252 

72598 

27402 

74405 

25595 

59 

2 

70779 

29221 

72628 

27372 

74435 

25565 

58 

3 

70810 

29190 

72659 

27341 

74405 

25535 

57 

4 

70841 

29159 

72689 

27311 

74494 

25.506 

56 

5 

70873 

29127 

72720 

27280 

74.-24 

25476 

55 

6 

70904 

29096 

72750 

27250 

74554 

25446 

54 

1 

70935 

29065 

72780 

27220 

74583 

25417 

53 

-  8 

70966 

29034 

72811 

27189 

74613 

25387 

52 

9 

70997 

29003 

72841 

27159 

74643 

25357 

51 

10 

9.71028 

10.28972 

9.72872 

10.27128 

9.74673 

10.25327 

50 

11 

71059 

28941 

72902 

27098 

74702 

25298 

49 

Vl 

71090 

28910 

72932 

27068 

74732 

25268 

48 

13 

71121 

28879 

72963 

27037 

74762 

2.5238 

47 

14 

71153 

28847 

72993 

27007 

74791 

25209 

46 

15 

71184 

28816 

73023 

26977 

74821 

25179 

45 

16 

71215 

28785 

73054 

26946 

74851 

25149 

44 

17 

71246 

28754 

73084 

26916 

74880 

25120 

43 

18- 

71277 

28723 

73114 

26886 

74910 

25090 

42 

19 

71308 

28692 

73144 

26856 

74939 

25061 

41  . 

20 

9.71339 

10.28661 

9.73175 

10.26825 

9.74969 

10.25031 

40 

21 

71370 

28630 

73-,'05 

26795 

74998 

25002 

39 

22 

71401 

28.599 

73235 

26765 

75028 

24972 

38 

23 

71431 

28569 

73265 

26735 

75058 

24942 

37 

24 

71462 

28538 

73295 

26705 

751)87 

24913 

36 

25 

71493 

28507 

73326 

26074 

75117 

24883 

35 

26 

71524 

28476 

7aS56 

26644 

75146 

24854 

34 

27 

71555 

28445 

733S6 

26614 

75176 

24824 

33 

28 

71586 

28414 

73416 

26584 

75205 

24795 

32 

29 

.   71617 

28383 

73446 

26554 

75235 

24765 

31 

30 

9.71648 

10.28352 

9.73476 

10.26524 

9.75264 

10.24736 

30 

31 

71679 

28321 

73507 

2G493 

75294 

24706 

29 

32 

71709 

28291 

73537 

26463 

75323 

24677 

28 

33 

717-10 

28260 

73567 

26433 

75353 

24647 

27 

34 

71771 

28229 

73597 

26403 

75882 

24618 

26 

35 

71802 

28198 

';3627 

26373 

75411 

24589 

25 

36 

71833 

28107 

73657 

26:343 

75441 

24559 

24 

37 

71863 

28137 

73687 

26313 

75470 

24530 

23 

38 

71894 

28105 

73717 

26283 

75500 

24500 

22 

39 

71925 

28075 

73747 

26253 

75529 

24471 

21 

40 

9.71955 

10.2S045 

9.73777 

10.26223 

9.75558 

10.24442 

20 

41 

71986 

28014 

73807 

26193 

75588 

24412 

19 

42 

72017 

27983 

73837 

26163 

7.5617 

1^383 

18 

43 

72048 

27952 

73867 

26133 

75647 

24353 

17 

44 

72078 

27922 

73897 

2610:^ 

75676 

24324 

16 

45 

72109 

27891 

73927 

26073 

75705 

24295 

15 

46 

72140 

27860 

73957 

26043 

75735 

24265 

14 

47 

72170 

27830 

73987 

26013 

75764 

24236 

13 

48 

72201 

27799 

74017 

25983 

75703 

24207 

12 

49 

72231 

27769 

74047 

25953 

75822 

24178 

11 

50 

9.72262 

10.27738 

9.74077 

10.2.5923 

9.758.52 

10.24148 

10 

51 

72293 

27707 

74107 

2.5893 

75881 

24119 

9 

52 

72323 

27677 

74137 

25863 

75910 

24090 

8 

53 

72354 

27646 

74166 

25834 

75939 

24061 

i 

54 

72384 

27616 

74196 

25804  . 

75969 

24031 

6 

55 

72415 

27585 

74226 

25774 

75098 

24002 

5 

56 

72445 

27555 

74256 

25744 

76027 

23973 

4 

57 

72476 

27524 

74286 

25714 

76056 

23944 

3 

58 

72506 

27494 

74316 

25684 

76086 

23914 

2 

59 

72537 

27463 

74345 

25655 

76115 

23885 

1 

60 

72567 

27433 

74375 

25625 

76144 

23856 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

"an 

62° 

61° 

60° 

TABLE  VIII.— LOG.  TANGENTS   AND  COTANGENTS.   263 


/ 

»0° 

31° 

S2° 

/ 

Tan 

Cotan 

Tan 

Cotau 

Tan 

Cotan 

0 

9.76144 

10  23856 

9.77877 

10.22123  ' 

9.79579 

10.20421 

60 

1 

70173 

23827 

77906 

22094 

79G07 

20393 

59 

o 

76-J02 

23798 

77935 

22065 

79035 

20365 

58 

3 

76231 

23709 

77963 

22037 

79663 

20337 

57 

4 

76J61 

23739 

77992 

22008 

79691 

20309 

56 

5 

76290 

23710 

78020 

21980 

79719 

20281 

55 

6 

76319 

23681 

78049 

21951 

79747 

20253 

54 

76348 

23652 

78077 

21923 

79776 

20224 

53 

8 

IVOI  1 

23623 

78106 

21894 

79804 

20196 

52 

9 

76406 

23591 

78135 

21865 

79882 

20168 

51 

10 

9.76435 

10.23505 

9.78163 

10.218.37 

9.79800 

10.20140 

50 

11 

76464 

23536 

78192 

21808 

79888 

20112 

49 

12 

76493 

23507 

78220 

21780 

79916 

20084 

48 

13 

76522 

23478 

78219 

21751 

79944 

20056 

47 

14 

765.->l 

23449 

78277 

21723 

79972 

20028 

46 

15 

765S0 

23420 

78306 

21694 

80000 

20000 

45 

16 

76609 

23391 

78334 

21666 

80028 

19972 

44 

17 

76639 

23361 

78363 

21637 

80056 

19944 

43 

18 

76668 

23332 

78391 

21609 

80084 

19916 

42 

19 

76697 

23303 

78419 

21581 

80112 

19888 

41 

20 

9.76725 

10.2.3275 

9.78448 

10.21552 

9.80140 

10.19860 

40 

21 

76754 

23246 

78476 

21524 

80168 

19832 

39 

22 

76783 

23217 

78505 

21495 

80195 

19805 

38 

28 

76812 

23188 

78533 

21467 

80223 

19777 

37 

24 

76841 

23159 

78562 

21438 

80251 

19749 

36 

25 

768';  0 

23130 

78590 

21410 

80279 

19721 

35 

26 

';6899 

23101 

78618 

21382 

80307 

19693 

34 

27 

16928 

2o072 

78647 

213.53 

80335 

19665 

33 

28 

76957 

23043 

78675 

21325 

80363 

19637 

32 

29 

76986 

23014 

78704 

21296 

80391 

19609 

31 

30 

9.77015 

10.22985 

9.787.32 

10.212G8 

9.80419 

10.19.581 

30 

31 

77044 

22956 

78760 

21240 

80447 

19.553 

29 

32 

77073 

22927 

78789 

21211 

80474 

19.526 

28 

33 

77101 

22899 

78817 

21183 

80.502 

19498 

27 

34 

77130 

22870 

78845 

21155 

80530 

19470 

26 

35 

771.59 

22841 

78874 

21126 

805.58 

19442 

25 

36 

77)88 

22812 

78902 

21098 

80586 

19414 

24 

37 

77217 

22783 

78930 

21070 

80614 

19386 

23 

38 

7^246 

22754 

78959 

21041 

80642 

19.358 

22 

39 

77274 

22726 

78987 

21013 

80669 

19331 

21 

40 

9.77303 

10.22697 

9.79015 

10.20985 

9.80897 

10.19.303 

20 

41 

77332 

22668 

7904:^ 

20957 

80725 

19275 

19 

42 

77361 

226;!9 

79072 

20928 

80753 

19247 

18 

43 

77390 

22610 

79100 

20900 

80781 

19219 

17 

44 

77'4I8 

22582 

79128 

20872 

80808 

19192 

16 

45 

77447 

22553 

79156 

20844 

80836 

19164 

15 

46 

77476 

22.524 

79185 

20815 

80864 

19136 

14 

47 

77505 

22495 

79213 

20787 

80892 

19108 

13 

48 

77-533 

22467 

79241 

20759 

80919 

19081 

12 

49 

77562 

22438 

79269 

20731 

80947 

19053 

11 

50 

9.77.591 

10.22409 

9.79297 

10.20703 

?. 80975 

10.19025 

10 

51 

77<;i9 

22381 

79326 

20674 

81003 

18997 

9 

52 

77648 

22352 

79354 

20646 

810.30 

18970 

8 

53 

77677 

22323 

79382 

20618 

81058 

18942 

< 

54 

77706 

22294 

79410 

20590 

81086 

18914 

6 

55 

77734 

22266 

79438 

20562 

81113 

18887 

5 

56 

77763 

22237 

79466 

20534 

81141 

18859 

4 

57 

77791 

22209 

79495 

20505 

81169 

18831 

3 

58 

77820 

22180 

79523 

20477 

81196 

18804 

2 

59 

77849 

22151 

79.551 

20449 

81224 

18776 

1 

60 

t 

22123 

79579 

20421 

81252 

18748 

0 

/ 

Tan 

Cotan 

Tan 

Cotnn 

Tan 

/ 

69° 

58° 

57° 

2G4  TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

33<» 

34° 

35° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.81252 

10.18748 

9.82899 

10.17101 

9.84523 

10.15477 

60 

1 

81279 

18721 

82926 

17074 

84550 

15450 

59 

1 

2 

81307 

18693 

82953 

17047 

84576 

15424 

58 

3 

81335 

18665 

829S0 

17020 

84603 

15397 

57 

4 

81362 

18638 

83008 

16992 

84630 

15370 

56 

5 

81390 

18610 

83035 

16965 

84657 

15343 

55 

6 

81418 

18582 

83062 

16938 

84684 

15316 

54 

7 

81445 

18555 

83089 

16911 

84711 

15289 

53 

1 

g 

81473 

18527 

83117 

16883 

84738 

15262 

52 

9 

81500 

18500 

83144 

16856 

84764 

15236 

51 

10 

9.81528 

10.18472 

9.83171 

10.16829 

9.84791 

10.15209 

50 

11 

81556 

18444 

83198 

16802 

84818 

15182 

49 

81583 

18417 

83225 

16775 

84845 

15155 

48 

18 

81611 

18389 

83252 

16748 

84872 

15128 

47 

14 

81638 

18362 

83280 

16720 

84899 

15101 

46 

81666 

18334 

83307 

16693 

84925 

15075 

45 

16 

81693 

18307 

83334 

16666 

84952 

15048 

44 

17 

81721 

18279 

83361 

10639 

84979 

15021 

43 

t  1 

18 

81748 

18252 

S^i'iH 

16612 

85006 

14994 

42 

19 

81776 

18224 

83415 

16585 

85033 

14967 

41 

20 

9.81803 

10.18197 

9.83142 

10.16558 

9.85059 

10.14941 

40 

21 

81831 

18169 

83470 

16530 

85086 

14914 

39 

22 

81858 

18142 

83497 

16503 

85113 

14887 

38 

23 

81886 

18114 

83524 

1&476 

85140 

14860 

37 

24 

81913 

18087 

83551 

16449 

85166 

14834 

36 

25 

81941 

18059 

83578 

16422 

85193 

14807 

35 

26 

81968 

18032 

83605 

16395 

85220 

14780 

34 

27 

81996 

18<X)4 

83632 

16368 

85247 

14753 

33 

^  1 

28 

82023 

17977 

53659 

16341 

85273 

14727 

32 

29 

82051 

17949 

83686 

16314 

85300 

14700 

31 

30 

9.82078 

10.17922 

9.83713 

10.16287 

9.85327 

10.14673 

30 

31 

82106 

17894 

83740 

16260 

85354 

14646 

29 

32 

82133 

17867 

83768 

16232 

853S0 

14620 

28 

S3 

82161 

17839 

83795 

16205 

85407 

14593 

27 

34 

82188 

17812 

83822 

16178 

85434 

14566 

26 

35 

82215 

17785 

83849 

16151 

85460 

14540 

25 

36 

82243 

17757 

83876 

16124 

85487 

14513 

24 

37 

82270 

17730 

83903 

16097 

85514 

14486 

23 

U  1 

38 
39 

82298 

17702 

83930 

16070 

85540 

14460 

22 

82325 

17675 

83957 

16043 

85567 

14433 

21 

40 

9.82352 

10.17648 

9.83984 

10.16016 

9.85594 

10.14406 

20 

41 

82380 

17620 

84011 

15989 

85620 

14380 

19 

42 

82407 

17593 

84038 

15962 

85647 

14353 

18 

43 

82435 

17565 

84065 

15935 

a5674 

14326 

17 

44 

82462 

17538 

84092 

15908 

85700 

14300 

16 

4.T 

824S9 

17511 

84119 

15881 

85727 

14273 

15 

46 

82517 

17483 

84146 

15854 

85754 

14246 

14 

47 

82544 

17456 

84173 

15827 

85780 

14220 

13 

48 

82571 

17429 

84200 

15800 

85807 

14193 

12 

49 

82599 

17401 

84227 

15773 

85834 

14166 

11 

^0 

9.82626 

10.17374 

9.84254 

10.15746 

9.85860 

10.14140 

10 

51 

82653 

17347' 

84280 

15720 

85887 

14113 

9 

52 

82681 

17319 

84^307 

15693 

85913 

14087 

8 

53 

82708 

17292 

84334 

1.5666 

85940 

14060 

i 

54 

82735 

17265 

84361 

15639 

85967 

14033 

6 

5.T 

82762 

17238 

84388 

15612 

85993 

14007 

5 

56 

82790 

17210 

84415 

1.5585 

86020 

13980 

4 

57 

82817 

17183 

84442 

1.5.558 

86046 

13954 

3 

»'  1 

58 

82844 

17156 

84469 

15.531 

86073 

13927 

2 

5Q 

82871 

17129 

^4496 

15.504 

86100 

13900 

1 

60 

82899 

17101 

84523 

15477 

86126 

13874 

0 
/ 

/ 

Coian 

Tan 

Cotan 

Tan 

Cotan 

Tan  i 

66° 

55° 

54° 

TABLE  Vlir.— LOG.  TANGENTS   AND   COTANGENTS.   2G5 


/ 

0 

36° 

37° 

38° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

9.86126 

10.13874 

9.87711 

10.12289 

9.89281 

10.10719 

60 

1 

86153 

13847 

87738 

12262 

89307 

10693 

59 

3 

86179 

13821 

87764 

12236 

89333 

10667 

58 

3 

86206 

13794 

87790 

12210 

89359 

10641 

57 

4 

80232 

13768 

87817 

12183 

89385 

10615 

56 

5 

86259 

13741 

87843 

12157 

89411 

10589 

55 

6 

86285 

13715 

87869 

12131 

89437 

10563 

54 

t 

86312 

13688 

87895 

12105 

89463 

10537 

53 

8 

86338 

13662 

87922 

12078 

89489 

10511 

52 

9 

86365 

13635 

87948 

12052 

89515 

10485 

51 

10 

9.86392 

10.13608 

9.87974 

10.12026 

9.89541 

10.10459 

50 

11 

86418 

13582 

88000 

12000 

89567 

10433 

49 

12 

86445 

13555 

88027 

11973 

89593 

10407 

48 

13 

86471 

13529 

88053 

11947 

89619 

10381 

47 

14 

86498 

13502 

88079 

11921 

89645 

10355 

46 

15 

86524 

13476 

88105 

11895 

89671 

10329 

45 

16 

86551 

13449 

88131 

11869 

89697 

10303 

44 

17 

86577 

13423 

88158 

11842 

89723 

10277 

43 

IS 

86603 

13397 

88184 

11816 

89749 

10251 

42 

19 

86630 

13370 

88210 

11790 

89775 

10225 

41 

20 

9.86656 

10.13344 

9.88236 

10.11764 

9.89801 

10.10199 

40 

21 

86683 

13317 

88262 

11738 

89827 

10173 

39 

86709 

13291 

88289 

11711 

89853 

10147 

38 

23 

86736 

13264 

88315 

11685 

89879 

10121 

37 

24 

86762 

13238 

88341 

11659 

89905 

10095 

36 

25 

86789 

13211 

8»367 

11633 

89931 

10069 

35 

26 

86815 

13185 

88393 

11607 

89957 

10043 

34 

27 

86842 

13158 

88420 

11580 

89983 

10017 

33 

28 

86868 

13132 

88446 

11554 

90009 

09991 

32 

29 

86894 

13106 

88472 

11.528 

90035 

09965 

31 

30 

9.86921 

10.13079 

9.88498 

10.11502 

9.90001 

10.09939 

30 

31 

86947 

13053 

88524 

11476 

90086 

09914 

29 

32 

86974 

13026 

88550 

11450 

90112 

09888 

28 

33 

87000 

13000 

88577 

11423 

90138 

09862 

27 

34 

87027 

12973 

88603 

11397 

90164 

09836 

26 

35 

87053 

12947 

8!<629 

11371 

90190 

09810 

25 

36 

87079 

12921 

88655 

11345 

90216 

09784 

24 

6 1 

87106 

12894 

88681 

11319 

90242 

09758 

23 

38 

87132 

12868 

88707 

11293 

90268 

09732 

22 

39 

87158 

12842 

88733 

11267 

90294 

09700 

21 

40 

9. 871 85 

10.12815 

9.88759 

10.11241 

9.90.320 

10.09680 

20 

41 

8^211 

12789 

88786 

11214 

90346 

09654 

19 

42 

87238 

12762 

88812 

11188 

90371 

09629 

18 

43 

87264 

12736 

88838 

11162 

90397 

09603 

17 

44 

87290 

12710 

88884 

11136 

90423 

09577 

16 

45 

87817 

12683 

88890 

11110 

90449 

09551 

15 

46 

87343 

12657 

88916 

11084 

90475 

09525 

14 

47 

87369 

12631 

88942 

11058 

90501 

09499 

13 

48 

87396 

12604 

88968 

11032 

90527 

09473 

12 

49 

87422 

12578 

88994 

11006 

90553 

09447 

11 

50 

9.87448 

10.12552 

9.89020 

10.10980 

9.90578 

10.09422 

10 

51 

87475 

12525 

89046 

10954 

90604 

09396 

9 

52 

87501 

12499 

89073 

10927 

90630 

0937Q 

8 

53 

87527 

12473 

89099 

10901 

90656 

09344 

i 

54 

87554 

]214ti 

89125 

10875 

90682 

09318 

6 

55 

87580 

12420 

89151 

10849 

90708 

09292 

5 

56 

87600 

12394 

89177 

10823 

90734 

09266 

4 

57 

87633 

12367 

89203 

10797 

90759 

09241 

3 

58 

87659 

12341 

89229 

10771 

90785 

09215 

2 

59 

87685 

12315 

89255 

10745 

90811 

09189 

1 

60 

/ 

87711 

122S9 

89281 

10719 

90837 

09163 

0 

Co  tan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

53° 

52° 

61° 

266   TABLE  Tin.— LOG.  TANGENTS    AND   COTANGENTS- 


/ 

39° 

40° 

41°       1 

60 

Tail 

Cotan ' 

Tau 

Cotan 

Tan 

Cotan  ! 

0 

9.90837 

10.09163 

9.92381 

10.07619 

9.9:3916 

10.0G084 

1 

90863 

09137 

92407 

07593 

93942 

06058 

59 

2 

90889 

09111 

92433 

07567 

9:3967 

060:33 

58 

3 

90914 

090S6 

92458 

07542 

93993 

06007 

57 

4 

90940 

09060 

92-184 

07516 

94018 

05982 

56 

5 

90966 

09034 

92510 

07490 

94044 

05956 

55 

6 

90992 

09008 

9-2535 

07465 

94069 

05931 

54 

7 

91018 

08982 

92561 

07439 

94095 

05905 

53 

8 

91043 

08957 

9-2587 

07413 

94120 

05880 

52 

9 

91069 

08931 

92612 

07388 

94146 

05854 

51 

10 

9.91095 

10.08905 

9.92638 

10.07:362 

9.94171 

10.058-29 

50 

11 

91121 

08879 

92663 

07:337 

94197 

05803 

49 

12 

91147 

0S853 

92689 

■  07311 

94222 

05778 

48 

13 

91172 

08828 

9-2715 

07285 

94248 

05752 

47 

14 

91198 

08802 

92740 

07260 

94-273 

05727 

46 

15 

91224 

08776 

92766 

07234 

94-299 

05701 

45 

16 

91250 

08750 

92792 

07208 

94324 

05676 

44 

17 

91276 

08724 

9-2817 

07183 

94350 

05650 

43' 

18 

91301 

08699 

92843 

07157 

94375 

056-25 

42 

19 

91327 

08673 

92868 

071  ;32 

94401 

05599 

41 

20 

9.91353 

10.08647 

9.9-2894 

10.07106 

9.944-26 

10.05.574 

40 

21 

91379 

08621 

92920 

07080 

944.52 

05548 

39 

22 

91404 

08596 

9-2945 

070.55 

94477 

055-23 

38 

23 

91430 

08570 

9-2971 

07029 

94503 

05497 

37 

24 

91456 

08544 

9-2996 

07004 

94528 

05472 

36 

25 

9148i 

08518 

9:3022 

06978 

94554 

05446 

:35 

26 

91507 

08493 

93048 

06952 

94570 

05421 

34 

27 

915^3 

08467 

93073 

06927 

94604 

05396 

33 

28 

91.559 

08441 

93099 

06901 

946:30 

05370 

32 

29 

9158.) 

08415 

93124 

06876 

94655 

05:345 

31 

30 

9.91610 

10.08390 

9.93150 

10.06850 

9.94681 

10.05319 

30 

31 

91636 

08364 

93175 

06825 

94706 

05294 

29 

32 

91662 

08338 

93201 

06799 

947:32 

05268 

28 

33 

9168S 

08312 

93227 

06773 

94757 

05243 

27 

34 

91713 

08287 

93-252 

06748 

94783 

05217 

26 

35 

91739 

08261 

93278 

06722 

94808 

05192 

25 

36 

91765 

08235 

9:3:303 

06697 

94834 

05166 

24 

37 

91791 

08209 

933-29 

06671 

94859 

05141 

23 

38 

91816 

081S4 

9:3354 

06646 

94884 

0.5116 

22 

39 

91842 

08158. 

933^0 

066-20 

94910 

05090 

21 

40 

9.91868 

10.081.32 

9.9:3406 

10.06594 

9.94935 

10.05065 

20 

41 

91893 

08107 

93431 

06569 

94961 

050:39 

19 

42 

91919 

08081 

93457 

06.54:3 

94986 

0.5014 

18 

48 

91945 

08055 

93482 

06518 

9.5012 

04988 

17 

44 

91971 

0S029 

9:3508 

06492 

95037 

04963 

16 

45 

91996 

08004 

93533 

06467 

95062 

049:38 

15 

46 

9-20-2-2 

07978 

93559 

06441 

95088 

04912 

14 

47 

92048 

07952 

93584 

0&416 

95113 

04887 

13 

48 

9-2073 

07927 

93610 

06390 

95139 

04861 

12 

49 

92099 

07901 

93636 

06:364 

95164 

04836 

11 

50 

9.92125 

10.07875 

9.93661 

10.06:339 

9.95190 

10.04810 

10 

51 

92150 

07850 

93687 

06313 

95215 

04785 

9 

52 

92176 

07824 

93712 

06288 

95240 

04760 

8 

L3 

92202 

07798 

93738 

06262 

95266 

04734 

1 

54 

92227 

07773 

93763 

06237 

95291 

04709 

6 

55 

92253 

07747 

93789 

06211 

95317 

046^3 

5 

56 

92279 

07721 

9:J814 

06186 

95342 

04658 

4 

57 

92304 

07696 

93840 

06160 

95:368 

046:32 

3 

58 

92330 

07670 

93865 

061:35 

95393 

04607 

2 

59 

92356 

07644 

93S91 

06109 

95418 

04582 

1 

60 

92381 

07619 

9:3916 

06084 

95444 

04556 

0 

1 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

50° 

49° 

48° 

TABLE  VIII.— LOG.  TANGENTS   AND   COTANGENTS. 


207 


/ 

42° 

43° 

44° 

/ 

Tan 

•  Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.95444 

10.04556 

9.96966 

10.03034 

9.98484 

10.01516 

60 

1 

95469 

04531 

96991 

03009 

98509 

014»1 

59 

o 

95495 

04505 

97016 

02984 

98534 

01466 

58 

3 

95520 

04480 

97042 

02958 

98560 

01440 

57 

4 

95545 

04455 

970G7 

02933 

98585 

01415 

56 

5 

95571 

04429 

97092 

02908 

98610 

01390 

55 

6 

95596 

04404 

97118 

02882 

98635 

01365 

54 

1 

95622 

0437.8 

97143 

02857 

98661 

01339 

53 

8 

95647 

04353 

97168 

02832 

98686 

01314 

52 

9 

95672 

04328 

97193 

02807 

98711 

01289 

51 

10 

9.95698 

10.04302 

9.97219 

10.02781 

9.98737 

10.01263 

50 

n 

95723 

04277 

97244 

02756 

98762 

01238 

49 

12 

95748 

04252 

97269 

02731 

98787 

01213 

48 

13 

95774 

04226 

97295 

02705 

98812 

01188 

47 

14 

95799 

04201 

97320 

02680 

98838 

01102 

46 

15 

95825 

04175 

97345 

02655 

98863 

01137 

45 

16 

95850 

04150 

97371 

02629 

98888 

01112 

44 

ir 

95875 

04125 

97396 

02604 

98913 

01087 

43 

18 

95901 

04099 

97421 

02579 

98939 

01061 

42 

19 

959V6 

04074 

97447 

02553 

98964 

01036 

41 

20 

9.95952 

10.04048 

9.97472 

10.02528 

9.98989 

10.01011 

40 

21 

95977 

04023 

97497 

02503 

99015 

00985 

39 

22 

96002 

03998 

97523 

02477 

99040 

00960 

38 

28 

96028 

03972 

97548 

02452 

99065 

00935 

37 

24 

96053 

03947 

97573 

02427 

99090 

00910 

36 

25 

96078 

03922 

97598 

02402 

99116 

00884 

35 

26 

96104 

03896 

97624 

02376 

99141 

00859 

34 

27 

96129 

03871 

97649 

02351 

99166 

00834 

33 

28 

96155 

03845 

97674 

02326 

99191 

00809 

32 

29 

96180 

03820 

97700 

02300 

99217 

00783 

31 

30 

9.96205 

10.03795 

9.97725 

10.02275 

9.99242 

10.00758 

30 

31 

96231 

03769 

97750 

02250 

99267 

00733 

29 

32 

96256 

03744 

97776 

02224 

99293 

00707 

28 

33 

96281 

03719 

97801 

02199 

99318 

00682 

27 

34 

96307 

03693 

97826 

02174 

99343 

00657 

26 

35 

96332 

03668 

97851 

02149 

99368 

00632 

25 

36 

96357 

03643 

97877 

02123 

99394 

00606 

24 

37 

96383 

03617 

97902 

02098 

99419 

OC581 

23 

38 

96408 

03592 

97927 

02073 

99444 

00556 

22 

39 

96433 

03567 

97953 

02047 

99469 

00531 

21 

40 

9.96459 

10.03541 

9.97978 

10.02022 

9.99495 

10.00505 

20 

41 

96484 

03516 

9S003 

01997 

99520 

00480 

19 

42 

96510 

03490 

98029 

01971 

99545 

00455 

18 

43 

96535 

03465 

98054 

01946 

99570 

00430 

17 

44 

96560 

03440 

98079 

01921 

99596 

00404 

16 

45 

96586 

03414 

98104 

01896 

99G21 

00379 

15 

46 

96611 

03389 

98130 

01870 

99646 

00354 

14 

47 

96636 

03364 

98155 

01845 

99672 

00328 

13 

48 

96662 

03338 

98180 

01 820 

99697 

00303 

12 

49 

96687 

03313 

98206 

01794 

99722 

00278 

11 

50 

9.96712 

10.03288 

9.98231 

10.01769 

9.99747 

10.00253 

10 

bi 

96738 

03262 

98256 

01744 

99773 

00227 

9 

52 

96763 

03237 

98281 

01719 

99798 

00202 

8 

53 

96788 

03212 

98307 

01693 

99823 

00177 

-•• 
1 

E4 

96814 

03186 

98332 

01668 

99848 

00152 

6 

55 

96839 

03161 

98857 

01643 

99874 

00126 

5 

56 

96H64 

03136 

98383 

01617 

99899 

00101 

4 

57 

96890 

03110 

98408 

01592 

99924 

0O076 

3 

58 

96915 

03085 

98433 

01567 

99949 

00051 

2 

59 

96940 

03060 

98458 

01542 

99975 

00025 

1 

GO 

96966 

03034 

98484 

01516 

10.00000 

00000 

0 

■ 
1 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

47° 

46» 

45° 

1 

1 

26S    A   FIELD-MAKUAL   FOR   RAILROAD    EKGIXEERS. 


TABLE  IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


The  Long  Chords,  ]\nd-Ordiuates,  Externals,  and  Tangent 
Distances  of  this  table  are  for  a  curve  of  5730  feet  radius.  To 
find  the  corresponding  functions  of  any  other  curve  divide  the 
tabular  values  by  the  degree  of  curve. 

For  metric  curves  having  20-metre  chords,  multiply  tne  degree 
by  5  and  enter  the  table  with  the  result  as  a  value  of  D,  the  tabu- 
lar values  being  taken  as  metres  instead  of  feet 

Thus  for  a  1^  30'  metric  curve  having  /=  45°  the  tangent  dis- 

tance  is  7"=  :p^V^  =  316.45  metres.     Again,  suppose  /=  38° 

and   the   long   chord  =  373.1  m.  known  and  D  required.     The 

3731.0 


tabular  L.  C.  is  3731  m. ;  therefore  Z>  = 


373.1  X  5 


2°0'. 


0 
2 

4 
6 
8 
10 
12 
14 
16 
18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 


0° 


L.  C. 

0.00 
3.. 33 
6.67 
10.00 
13.33 
16.67 
20.00 
23.33 
26.67 
30.00 

33.33 
36.67 
40.00 
43.33 
46.67 
50.00 
53.33 
56.67 
60.00 
63.33 

66.67 
70.00 
73.33 
76.67 
80.00 
83.33 
86.67 
90.00 
93  33 
96.67 
100.00 


M. 


E. 


0.000 
0.000 
0.001 
0.002 
0.004 
0.006 
0.009 
0.012 
0.015 
0.019 

0.024 
0.029 
0.035 
0.041 
0.048 
0.054 
0  002 
0  070 
0.079 
0.088 

0.097 
0.107 
0.117 
0.128 
0.140 
0.151 
0.164 
0  176 
0.190 
0.204 
0.218 


0.000 
0.000 
0.001 
0.002 
0.004 
0.006 
0.009 
0.012 
0.015 
0.019 

0.024 
0.029 
0.035 
0.041 
0.048 
0.054 
0.062 
O.OfO 
0.079 
0.088 

0.097 
0.107 
0.117 
0.128 
0.140 
0.151 
0.164 
0.170 
0.190 
0.204 
0.218 


0.00 

1.67 

3.33 

5.00 

6.67 

8.33 

10.00 

11.67 

13.33 

15.00 

16.67 
18.33 
20.00 
21.67 
23.33 
25.00 
26.67 
28.33 
30.00 
31.67 

33  33 

35.00 

36.  nr 

38.. 33 
40.00 

41.  or 

43.. 33 
45  0(» 

■1(1.  t;: 

4,s  3.-! 
•5(1  (»0 


1° 

L.  C. 

M. 

E. 

T. 

100.00 

0.218 

0.218 

50.00 

0 

103.33 

0.233 

0.233 

51.67 

o 

100.66 

0.248 

0.248 

53.33 

4 

110.00 

0.264 

0.264 

55.00 

6 

113.33 

0.280 

0.280 

56.67 

8 

116.66 

0.297 

0.297 

58.33 

10 

120  00 

0.314 

0.314 

60.00 

12 

123.33 

0.332 

0.332 

61.67 

14 

12G.66 

0.3.50 

0..350 

63.33 

16 

130.00 

0.368 

0.368 

65.00 

18 

1.33.33 

0.388 

0.388 

66.67 

20 

130.66 

0.407 

0.407 

68.33 

22 

140.00 

0.427 

0.427 

70.00 

24 

143.33 

0.448 

0.448 

71.67 

26 

146.66 

0.4G9 

0.4G9 

73.33 

28 

150  00 

0.491 

0.491 

75.00 

30 

153.33 

0..513 

0.513 

76.67 

32 

156.66 

0.536 

0.536 

78.33 

34 

160.00 

0.5.59 

0,559 

80.00 

36 

163.33 

0.582 

0.582 

81.67 

38 

166.66 

0.606 

0.606 

83.. 33 

40 

170.00 

0.630 

0.630 

85.00 

42 

173,33 

0.655 

0.655 

86.67 

44 

17G.66 

0.681 

0.681 

88.33 

46 

180.00 

0.706 

0.706 

90.00 

48 

183.33 

0  733 

0.733 

91.67 

50 

186.66 

0.760 

0.760 

93.33 

52 

190.  (K) 

0.7SS 

0  788 

95.00 

54 

193.33 

0.815 

0.815 

96.67 

56 

196  66 

0.844 

0.844 

98.33 

58 

199.98 

0  873 

0  873 

100.00 

60  ( 

IX.— FUNCTIONS   OF    A    ONE-DEGREE   CURVE.     2fi9 


t 

k>0 

3 

o 

/ 

L.  0. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

199.98 

0.873 

0.S73 

100.00 

299.96 

1.964 

1.964 

1,50.07 

0 

2 

203.31 

0.902 

0.902 

101.67 

303.29 

2.008 

2.009 

151.74 

2 

4 

206.64 

0.9.32 

0.932 

103.34 

306.62 

2.053 

2.054 

1,53.41 

4 

6 

209.97 

0.962 

0.9G2 

105.01 

309  95 

2.098 

2.099 

1.55.08 

6 

8 

213.31 

0.993 

0.993 

106.68 

313.29 

2. 143 

2.144 

156.75 

8 

10 

210.64 

1.024 

1.024 

108.35 

316.62 

2.188 

2.189 

158.42 

10 

Vi 

219.97 

1.0.56 

1.0.56 

110.02 

319.95 

2.235 

2.2.36 

160.09 

12 

14 

223.30 

1.088 

1.088 

111.69 

323.28 

2.282 

2.283 

161.76 

14 

16 

226.64 

1.121 

1.121 

113.36 

326.62 

2. 329 

2.3,30 

163.43 

16 

18 

229.97 

1.154 

1.154 

115.02 

329.95 

2.-376 

2.. 377 

165.09 

18 

20 

233.30 

1.188 

1.188 

116.69 

333.28 

2.424 

2.425 

166.76 

20 

22 

236.63 

1.222 

1.222 

118.36 

336  61 

2.473 

2.474 

168.43 

22 

24 

239.97 

1.256 

1.256 

120.03 

339.95 

2.523 

2.523 

170.10 

24 

26 

243.30 

1.292 

1.292 

121.70 

343.28 

2. 572 

2.573 

171.77 

26 

28 

246.63 

1.328 

1.328 

123.. 37 

346.61 

2.622 

2.623 

173  44 

28 

30 

249.96 

1.364 

1.364 

125.03 

349.94 

2.672 

2.673 

175.10 

30 

;i2 

253.29 

1.399 

1.399 

126.70 

3.53  27 

2.724 

2.725 

176.72 

32 

34 

2.56.62 

1.437 

1.4.37 

12S  37 

3.")6 .  60 

2.776 

2.777 

178.39 

34 

36 

259.96 

1.475 

1.475 

130.04 

359.94 

2.828 

2.829 

180.06 

36 

38 

263.29 

1.513 

1.513 

131.71 

363.27 

2.880 

2.881 

181.73 

38 

40 

266.62 

1.552 

1..552 

133  38 

366  60 

2  933 

2.9.34 

183.40 

40 

42 

269.96 

1.592 

1..592 

135.05 

369.94 

2.987 

2.988 

185.07 

42 

44 

273.29 

1.632 

1.632 

136.72 

.373  27 

3.042 

3  043 

186.74 

44 

46 

276.62 

1.672 

1.672 

138.. 38 

376.60 

3.096 

3  097 

188.40 

46 

48' 

279.96 

1.712 

1.712 

140.05 

379.94 

3.151 

3.1.52 

190.07 

48 

50 

283.29 

1.752 

1.7.52 

141.72 

383.27 

3.206 

3.207 

191.74 

50 

52 

286.62 

1.794 

1.794 

143  39 

.386  60 

3  263 

3.264 

193.41 

52 

54 

289.96 

1.836 

1.836 

145.06 

389  94 

3.. 320 

3.321 

195.08 

54 

56 

293  29 

1.878 

1.878 

146.73 

393.27 

3.. 377 

3  378 

196.75 

56 

58 

296.02 

1.921 

1.921 

148.40- 

3'.»t; .  60 

3.434 

3.435 

198.42 

58 

60 

299.96 

1.964 

1.964 

1.50  07 

399.94 

3.491 

3.492 

2u0  09 

60 

4< 

> 

5 

3 

/ 

t 

L.  C. 

M. 

E. 

T. 

L.  C 

M. 

E. 

T. 

0 

.399.94 

3.491 

3.492 

200.09 

499.88 

5.4,54 

5.4.59 

250.17 

0 

o 

403.27 

8.5.50 

3.. 551 

201.76 

503.21 

5.. ^27 

5.033 

251.84 

2 

4 

406.60 

3.609 

3.610 

203.43 

506  54 

5.601 

5.607 

253.51 

4 

6 

409.93 

3.668 

3.670 

205.10 

509.87 

5.675 

5.681 

255.18 

6 

8 

413.26 

3.727 

3.730 

206.77 

513.20 

5.749 

5.7.55 

256.85 

8 

10 

416.59 

3.787 

3.790 

208.44 

516  .53 

5.823 

5.829 

258.. 52 

10 

12 

419  92 

3.848 

3.851 

210.11 

519.86 

5.899 

5.905 

260.20 

12 

14 

423.26 

3.910 

3  913 

211.77 

523.19 

5.975 

5.981 

261.86 

14 

16 

426.59 

3.972 

3.975 

213.45 

526.. 52 

6.0.52 

6.058 

263.54 

16 

18 

429.92 

4.034 

4.037 

215.11 

529.85 

6.129 

6.1-35 

265.20 

18 

20 

433.25 

4.096 

4.099 

216.78 

533.18 

6.206 

6.212 

266.87 

20 

22 

436.58 

4.160 

4.163 

218.45 

536  51 

6.284 

6.290 

268.54 

22 

24 

439.91 

4.224 

4.227 

220.12 

539.84 

6.362 

6.369 

270.21 

24 

26 

443.24 

4.288 

4.291 

221.79 

543.17 

6.441 

6.448 

271.88 

26 

28 

446.. 58 

4.353 

4.3.56 

223.46 

546.. 50 

6.520 

6.527 

273.. 54 

28 

30 

449.91 

4.418 

4.421 

225.13 

549.83 

6.599 

6.606 

275.21 

30 

32 

4.53.24 

4.484 

4.467 

226  80 

553  17 

6.680 

6.687 

276.88 

32 

34 

456.57 

4.550 

4.554 

228.47 

556.50 

6.761 

6.768 

278.55 

34 

36 

459.90 

4.617 

4.621 

2.30,14 

559.83 

6.842 

6.849 

280.23 

36 

38 

463.23 

4.684 

4.688 

231.81 

563,16 

6.923 

6.931 

281.90 

38 

40 

466.56 

4.751 

4.755 

233  48 

566.49 

7.005 

7  013 

283.57 

40 

42 

469.89 

4.820 

4.824 

235.15 

569.82 

7.088 

7.096 

285.24 

42 

44 

473.23 

4.889 

^.893 

2.36.82 

573.15 

7  171 

7.180 

286.91 

44 

46 

476  .56 

4.958 

4  962 

2.38.48 

576.48 

7  255 

7.264 

288.59 

46 

48 

479.89 

5.027 

5.031 

240.15 

579.81 

7.339 

7.348 

290.26 

48 

50 

483.22 

5.096 

5.100 

241.82 

583.14 

7  423 

7.432 

291.93 

50 

52 

486.55 

5.167 

5.171 

243.49 

586.47 

7.508 

7.517 

293.60 

52 

54 

489  88 

5.238 

5.243 

245  16 

.589.80 

7. 593 

7.603 

295.27 

54 

56 

493.21 

5.310 

5.315 

216  83 

593.13 

7  678 

7  689 

296  95 

56 

58 

496  .54 

5,. 382 

5  3S7 

248  ,50 

,596.46 

7.764 

7  775 

298  62 

58 

60 

499.88 

5  454 

5.4.59 

2."0  17 

599. HO 

7.8.50 

7  861 

300.30 

60 

2?0     IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


t 

t° 

JO 

/ 

L   C. 

M. 

E 

T. 

L.  C. 

M. 

E. 

T. 

1 

0 

599.80 

7.850 

7.861 

300.30 

699.60 

10.69 

10.71 

350.44 

0 

2 

603.13 

7.940 

7.951 

301 .97 

702.93 

10.79 

10.81 

3.52.11 

2 

4 

606.46 

8.030 

8.041 

303.64 

706.26 

10,90 

10.92 

353.79 

4 

6 

609.78 

8.120 

8.131 

305.81 

709,. 58 

11.00 

11.02 

355.46 

6 

8 

613.11 

8.210 

8.2J1 

306.98 

712.91 

11.11 

11.13 

357.13 

8 

10 

616.44 

8.300 

8.311 

308.65 

716.24 

11.21 

11.23 

358.81 

10 

12 

619.76 

8.390 

8.401 

310.32 

719  .56 

11.31 

11.33 

360.48 

12 

14 

623.09 

8.480 

8.491 

311.99 

722.89 

11.42 

11.44 

362.15 

14 

16 

626.42 

8.. 570 

8.581 

;il3.66 

726  21 

11,52 

11.54 

363  83 

16 

IS 

629.74 

8.660 

8.671 

315.33 

729.. 53 

11.63 

11.65 

365.50 

18 

•20 

633.07 

8.750 

8.761 

317.00 

732.86 

1 1 .  73 

11.75 

367,17 

20 

22 

636.40 

8.844 

8.856 

318.67 

736.19 

11.84 

11.86 

368.85 

22 

24 

639.72 

8.939 

8.951 

320.34 

739  51 

11.95 

11.97 

370,52 

24 

26 

643.05 

9.033 

9.046 

322  01 

742.84 

12.06 

12.08 

372.19 

26 

28 

646.38 

9.128 

9.141 

323.68 

746.17 

12.17 

12.19 

373.86 

28 

30 

649.70 

9.222 

9.236 

325.35 

749.49 

12.27 

12.. 30 

375.54 

30 

32 

653.03 

9.317 

9. 331 

327.02 

7.52.82 

12.38 

12.41 

377.22 

32 

34 

656.36 

9.411 

9.4-.'6 

328.69 

756.15 

12.49 

12  52 

378.89 

34 

36 

659.69 

9.506 

9.521 

33U.37 

7.59.47 

12.60 

12.63 

380  57 

36 

38 

663.02 

9.600 

9.616 

332  04 

762.80 

12.71 

12.74 

382.24 

38 

40 

666.34 

9.695 

9.712 

3.3-3  71 

766.13 

12.82 

12.85 

383.92 

40 

42 

669.67 

9.794 

9.812 

335.38 

769.45 

12,93 

12.96 

385.60 

42 

44 

673.00 

9.894 

9.912 

337.05 

< /2. (0 

13,04 

13.08 

387.27 

44 

46 

676.32 

9.993 

10.01 

338. '13 

776.11 

13.15 

13.19 

388.95 

46 

48 

679.65 

10.09 

10.11 

340.40 

779.43 

13  26 

13.31 

390.62 

48 

50 

682.98 

10.19 

10.21 

342.07 

782.76 

13,37 

13.42 

392.30 

50 

52 

686.30 

10.29 

10.31 

343.74 

786.09 

13.48 

13.53 

393.98 

52 

54 

689.63 

10. .39 

10.41 

345.41 

789.41 

13  .59 

13.65 

395.65 

54 

56 

692.96 

10  49 

10.51 

347.08 

792.74 

13.70 

13.76 

397.33 

56 

58 

696.28 

10.59 

10,61 

348.76 

796.07 

13.81 

13.88 

399.01 

58 

60 

699.60 

10.69 

10.71 

3.50.44 

799  40 

13  96 

13.99 

400.70 

60 

/ 

8 

a 

8 

1° 

/ 

L.  C. 

799.40 

M. 
13.96 

E. 
13.99 

T. 
400.70 

L.  C. 

M. 

E. 

T. 

0 

899.10 

17.66 

17.71 

450.95 

0 

2 

802.72 

14.07 

14.10 

402  37 

902.42 

17.79 

17.84 

452.63 

2 

4 

806.04 

14.19 

14,22 

404.05 

905,74 

17,92 

17.98 

454.31 

4 

6 

809.37 

14.31 

14  34 

405.72 

909.07 

18.06 

18.11 

455.98 

6 

8 

812.69 

14  43 

14.46 

407.39 

91 2. 39 

18.19 

18.25 

457.66 

8 

10 

816.01 

14.. 55 

14.58 

409.06 

915.71 

18.32 

18.38 

4.59,34 

10 

12 

819.34 

14.66 

14.70 

410.74 

919.04 

18.46 

18.. 52 

461.02 

12 

14 

822.66 

14.78 

14.82 

412.41 

922.. 36 

18.59 

18.65 

462.70 

14 

16 

825.98 

14.90 

14.94 

414  OS 

925.68 

18.72 

18.79 

464,37 

16 

18 

829.31 

15.02 

15.06 

415.75 

929.01 

18,86 

18.92 

466.05 

18 

20 

832.63 

15.14 

15.18 

417.43 

932  .33 

18.99 

19.06 

467.73 

20 

22 

835.95 

15.26 

15.30 

419.10 

935.65 

19.12 

19.19 

469,41 

22 

24 

839.28 

15.38 

15.43 

4-:o  77 

9.38.98 

19.26 

19.33 

471.08 

24 

26 

842.60 

15.51 

15.. 55 

422  45 

942.. 30 

19.40 

19.47 

472.76 

26 

28 

845.92 

15.63 

15.68 

424   12 

945.62 

19.54 

19.61 

474.43 

28 

30 

849.25 

15.75 

15.80 

425.79 

948.95 

19.68 

19.75 

476.10 

30 

32 

852.57 

15.88 

15  93 

427.47 

9.52.27 

19.82 

19.89 

477.78 

32 

34 

8.-)5.89 

16.00 

16.05 

429.15 

955.59 

19.96 

20.03 

479.46 

.34 

36 

859.22 

16.12 

16.18 

4.30.82 

9.58.92 

20.10 

20.17 

481.14 

36 

38 

862.54 

16.25 

16.30 

432.. 50 

962.24 

20.24 

20.31 

482.83 

38 

40 

865.86 

16.38 

16.43 

434.13 

965.56 

20.38 

20.45 

484.51 

40 

42 

869.19 

16.50 

16.55 

435.86 

968.89 

20  .52 

20.59 

486.19 

42 

44 

872.51 

16.63 

16.68 

437.54 

972  21 

20.66 

20.74 

487.87 

44 

46 

875.83 

16.76 

16.81 

439.2! 

975.. 53 

20.80 

20.88 

489.55 

46 

48 

879.16 

16.89 

16.94 

440.89 

978  86 

20.94 

21.03 

491.24 

48 

50 

882.48 

17.02 

17.07 

4  42.. 57 

9S2.18 

21.09 

21.17 

492,92 

50 

52 

885.80 

17.14 

17.19 

444  25 

985.50 

21.23 

21,31 

494.60 

52 

54 

8S9.13 

17.27 

17  32 

445.93 

988.83 

21.. 37 

21.40 

496,28 

54 

56 

892.45 

17.40 

17.45 

447.60 

992.15 

21.51 

21,60 

497.96 

56 

58 

895.77 

17.. 53 

17  .58 

449.28 

995  47 

21.6.T 

21.75 

499,65 

58 

60 

899.10 

17.66 

17.71 

4.50.95 

998.80 

21.80 

21,89 

501.32 

60 

IX.     FUJSCTIONS  OF  A  ONE-DEGREE  CURVE. 

27i 

i 

10°                     1 

11° 

/ 

/ 

L  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

1  -»  -^ 

0 

998.8 

21.80 

21.89 

501.32 

1098.4 

26.38 

26.50 

551.74 

0 

o 

1002.1 

21.94 

22.03 

.503.00 

1101.7 

26.. 54 

26.66 

553.42 

o 

4 

1005.4 

22.09 

22.18 

.504.68 

1105.0 

26.70 

26.83 

555.10 

4 

6 

1008.8 

22.24 

22.33 

.506.. 36 

1108  3 

26.86 

26.99 

556.78 

6 

8 

1012.1 

22.39 

22.48 

508.04 

1111.7 

27.02 

27.16 

558.46 

8 

10 

1015.4 

22.54 

22.63 

509.72 

1115.0 

27.19 

27.. 32 

560.14 

10 

U' 

1018.7 

22.68 

22.78 

511.40 

1118.3 

27.35 

27.48 

561.82 

12 

14 

1022.0 

32.83 

22.93 

513.08 

1121.6 

27.51 

,27.65 

563.50 

14 

16 

1025.4 

22.98 

23.08 

514.76 

1124.9 

27.67 

27.81 

565.18 

16 

18 

1028.7 

23.13 

23  23 

516.44 

1128.2 

27.83 

27.98 

566.86 

18 

20 

1032.0 

23.28 

23.38 

518  12 

1131.6 

28.00 

28.14 

568.54 

20 

22 

1035.3 

23.43 

23.. 53 

519.80 

1134.9 

2g.l7 

28.30 

570.22 

22 

24 

1038.6 

23.58 

23  68 

521.48 

1138.2 

28.34 

28.47 

.571.90 

24 

26 

1042.0 

23.73 

23.84 

.523  16 

1141.5 

28.. 50 

28.64 

573.58 

26 

28 

1045  3 

23.88 

23.99 

.524.85 

1144.8 

28.67 

28.81 

575  27 

28 

30 

1048.6 

24.04 

24.14 

526., 53 

1148.1 

28.84 

28.98 

.576,95 

30 

32 

1051.9 

24.19 

24.30 

.528.21 

1151.5 

29.00 

29.14 

578.63 

32 

34 

1055.2 

24.34 

24.45 

529.89 

1154.8 

29.17 

29.31 

.580.32 

34 

36 

1058.6 

24.49 

24.60 

531.57 

11.58.1 

29.34 

29.48 

582.00 

36 

38 

1061.9 

24.64 

24.76 

533.25 

1161.4 

29.. 50 

29.65 

583.69 

38 

40 

1065.2 

24.80 

24.91 

534.93 

1164.7 

29.67 

29.82 

585.37 

40 

42 

1068.5 

24.95 

25.06 

536.61 

1168.0 

29.84 

29.99 

587.05 

42 

44 

1071.8 

25.11 

25.22 

538.29 

1171.4 

30.01 

30.17 

588.74 

44 

46 

1075.2 

25.27 

25.38 

.539.97 

1174.7 

.30.18 

30.34 

590.42 

46 

48 

1078.5 

25.43 

25.54 

541.65 

1178.0 

30.35 

30.52 

592.11 

48 

50 

1081.8 

25.59 

25.70 

543., 33 

1181.3 

30. 53 

30.69 

593.79 

50 

52 

1085.1 

25.74 

25.86 

545.01 

1184.6 

.30  70 

30.86 

595.47 

52 

54 

1088.4 

25.90 

26.02 

546.69 

1187.9 

.30.87 

31.04 

597.16 

54 

56 

1091.8 

26.06 

26.18 

54S..S7 

1191.3 

31.04 

31.21 

598.84 

56 

58 

1095.1 

26.22 

26.34 

.5.50  06 

1H)4.6 

31.21 

31.39 

600.53 

58 

60 

1098.4 

26.38 

26.. 50 

551.74 

1107.9 

31.39 

31.56 

602.22 

60 

3 

12°                          1 

13° 

/ 

L.  C. 

31. 

E. 

T. 

L.  C. 

M. 

E 

T. 

1197.9 

31.. 39 

31.. 56 

602.22 

1207.3 

36.83 

37.07 

652.87 

0 

2 

1201.2 

31.57 

31.  IS 

603.91 

1300.6 

37.02 

37.26 

6.54.56 

o 

4 

1204.5 

31.74 

31.01 

605.00 

1303.9 

37.21 

37.46 

6,56.25 

4 

6 

1207.8 

31.92 

32.09 

607.28 

1307.2 

37  40 

37.65 

657.93 

6 

8 

1211   1 

32  09 

32.27 

608.97 

1310.5 

37.59 

37.85 

650.62 

8 

10 

1214.5 

32.27 

32.45 

610.66 

1.313.8 

37  79 

38.04 

661.31 

10 

12 

1217.8 

32.45 

32.63 

612.35 

1317.2 

37.98 

38.:i3 

663.00 

12 

14 

1221.1 

32.62 

32.81 

614.04 

1320.5 

38.17 

38.43 

664.69 

14 

16 

1224.4 

32.80 

32.99 

615.72 

1323.8 

38.36 

38.02 

666.37 

16 

18 

1227.7 

32.97 

33.17 

617.41 

1327.1 

38.55 

38.82 

668.06 

18 

20 

1231.0 

33.15 

33.35 

619.10 

1330.4 

38.75 

39.01 

669.75 

20 

22 

1231.3 

33.33 

33.53 

6-.I0.79 

1333.7 

38.95 

39.20 

671.44 

22 

24 

1237.7 

.33.51 

33.72 

622.48 

1337.0 

39.15 

39.40 

673.13 

24 

26 

1241.0 

33.69 

.33.90 

624.16 

1340.3 

.39.35 

39.60 

674.81 

26 

28 

1244.3 

33.87 

34.09 

6-'5.85 

1343.6 

39.54 

39.80 

676.51 

28 

30 

1247  6 

34.06 

34.27 

627.. 55 

1346.9 

39.74 

40.00 

678.20 

30 

32 

12.50.9 

34.24 

34.45 

629.24 

13.50.3 

.39.94 

40.19 

679.89 

32 

34 

1254.2 

34.42 

.34  64 

630  93 

13.53.6 

40.13 

40.. 39 

681.58 

34 

36 

1257.5 

.34.60 

34.82 

632.61 

13.56.9 

40.33 

40  .59 

683.26 

36 

38 

1260.8 

34.78 

35.01 

634.30 

1360.2 

40.52 

40.79 

684.95 

38 

40 

1264.2 

34  97 

35.19 

6.35.99 

1363.5 

40.71 

40.99 

686.64 

40 

42 

1267.5 

35.16 

35.37 

637.08 

1366.8 

40.91 

41.19 

688.33 

42 

44 

1270.8 

35.34 

35.56 

639.37 

1370  1 

41.11 

41.40 

690.02 

44 

46 

1274.1 

35.. 53 

35.75 

641.05 

1373.4 

41.31 

41.60 

691.70 

46 

48 

1277.4 

35.71 

35.94 

642.74 

1376.7 

41.51 

41.81 

693.. 39 

48 

50 

1280.7 

35.90 

36.13 

644.43 

1380.0 

41.71 

42.01 

695.08 

50 

52 

1284.0 

36.09 

36  31 

646.12 

1383.4 

41.91 

42.21 

690.77 

52 

54 

1287.4 

36.27 

3(i..50 

647.81 

1380.7 

42  11 

42.42 

698.46 

54 

56 

1290.7 

36.46 

30 .  69 

649.49 

1300.0 

42.31 

42.02 

700.14 

56 

58 

1294.0 

36.64 

.36.88 

651.18 

1393.3 

42.51 

42.83 

701.83 

58 

60 

1297.3 

36 .  K3 

3r.07 

6.52.87 

1  1300.6 

42  71 

43.03 

703.. ^:3 

60     1 

272     IX.— FUNCTIONS   OF   A   ONE-DEGREE 


CURVE. 


14" 

15° 

f 

L.  C. 

1306.6 

M. 

42.71 

E. 
43.03 

T. 

r03.53 

L.  C. 

M.         E. 

T. 

/ 

0 

1495.9 

49.02    49.44 

754.35 

0 

2 

1399.9 

42.92 

43.23 
43.44 

705.23 

1409.2 

49.24    49.66 

756.05 

o 

4 

1403.2 

43.12 

706.92 

'1502.5 

49.40    49.89 

757.74 

4 

6 

1406.5 

43.33 

43.65 

708.62 

1505.8 

49.08    50.11 

759.44 

6 

8 

1409.8 

43.53 

43.86 

710.31 

1500.1 

49  90    50.34 

761.13 

8 

10 

1413.1 

43.74 

44.07 

712.01 

1512.4 

50.12    50.56 

762.83 

10 

12 

1416.5 

43.94 

44.28 

713.71 

1515.7 

50.34    50.78 

764.53 

12 

14 

1419.8 

44.15 

44.49 

715.40 

1519.0 

50.56    51.01 

766.22 

14 

16 

1423.1 

44.35 

44.70 

717.10 

1522.3 

50  78    51.23 

767.92 

16 

18 

1426.4 

44.56 

44.91 

718.79 

1525.6 

51.00    51.46 

769.61 

18 

20 

1429.7 

44.77 

45.12 

720.49 

1528.9 

51.22    51.68 

771.31 

20 

1433.0 

44.98 

45.33 

722.20 

1532.2 

51.44    51.90 

773.01 

22 

24 

1436.3 

45.19 

45.54 

723.89 

1535.5 

51.67    52.13 

774.70 

24 

26 

1439.6 

45.40 

45.76 

725.59 

1538.8 

51.89    52.36 

776.40 

26 

28 

1442.9 

45.61 

45.97 

727.28 

1542.1 

52.12    .52.59 

778.09 

28 

30 

1446.2 

45.82 

46.18 

72S.i»7 

1545.4 

52.34     52.82 

779.79 

30 

32 

1449.6 

46.03 

46.40 

730.06 

1548.7 

52. 57    53.05 

781 .49 

32 

34 

1452.9 

46.24 

46.61 

732  35 

15.52.0 

52.79    .53.28 

783.19 

34 

36 

1456.2 

46.45 

46.82 

734.05 

1555.3 

53.02    53.51 

784.89 

36 

38 

1459.5 

46.66 

47.04 

735.74 

1558.6 

53.24    53.74 

786.59 

38 

40 

1462.8 

46.87 

47.25 

737.43 

1561.9 

53.47    53.97 

788.29 

40 

42 

1460.1 

47.  as 

47.46 

739.12 

1565.2 

53.69    54.20 

789.99 

42 

44 

1469.4 

47.30 

47.68 

740. SI 

1508.5 

53.92    54.44 

791.69 

44 

46 

1472.7 

47.51 

47.90 

742.51 

1571.8 

54.15    54.67 

793.39 

46 

48 

147C.0 

47.73 

48.12 

744.20 

1575.1 

54.38    54.91 

795.09 

48 

50 

1479.3 

47.94 

48.34 

745.89 

15T8.4 

54.61     55.14 

796.79 

50 

52 

1482.7 

48.16 

48.56 

747.58 

1581.7 

54.84    55.37 

798.49 

52 

54 

1486.0 

48.37 

48.78 

749.27 

1585.0 

■55.07     55.61 

800.19 

54 

56 

1489.3 

48.59 

49.00 

750.97 

15S8.3 

55.30    55.84 

801.89 

56 

58 

1492.6 

48.80 

49.22 

752.66 

1591.6 

55  53    56.08 

803.59 

58 

60 

1495.9 

49.02 

49.44 

754.35 

1594.9 

55.76    56.31 

805.29 

60 

/ 

16°                            1 

17"                           , 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M.          E. 

T. 

0 

1594.9 

55.76 

56.31 

805.29 

1693.9 

62  94    63.64 

856.35 

0 

o 

1598.2 

55.99 

56.. 54 

806.99 

1697.2 

63.18    63.89 

858.05 

2 

4 

1601.5 

56.23 

56.78 

808.64 

1700.5 

63.43    64.15 

859.76 

4 

C 

1604.8 

56.46 

57.02 

810.39 

1703.8 

63.08     64.40 

801.46 

6 

8 

1608.1 

56.70 

57.26 

812.09 

1707.1 

63  93    64.66 

863.16 

8 

10 

1611.4 

56-93 

57.50 

813.79 

1710.4 

64.18    64.91 

864.87 

10 

12 

1614.7 

57.17 

57.74 

815.49 

1713.7 

64.42    65  16 

866.57 

12 

14 

1618.0 

57.40 

57.98 

817.19 

1710.9 

64.67    65.42 

868.27 

14 

16 

1621.3 

57.64 

58.22 

818.89 

1720.2 

64.92    65  67 

869.98 

16 

18 

1624.6 

57.87 

58.46 

820.59 

1723.5 

65.17    65.93 

871.68 

18 

20 

1627.9 

58.11 

58  TO 

822.29 

1726.8 

65.42    66.18 

873.38 

20 

22 

1631.2 

58.34 

58.94 

823.09 

1730.1 

65.67    66.43 

875.09 

22 

24 

1634.5 

58.58 

59  19 

825.09 

17:^:3.4 

65.93    66.69 

876.79 

24 

26 

1637.8 

58.82 

59.43 

827  39 

17:^6.7 

66. Ig    66.95 

878.49 

26 

28 

1641.1 

59.06 

59.68 

829.09 

1740.0 

66.44    67.21 

880.20 

28 

30 

1644.4 

59.30 

59  92 

830  79 

1743.3 

66.69    67.47 

881.90 

30 

32 

1647.7 

59.54 

60.16 

832.49 

1746.6 

66.94    67.72 

883.61 

32 

34 

1651.0 

59.78 

60.41 

834.20 

1749.9 

67.20    67.98 

885.32 

34 

36 

1654.3 

60.02 

60.65 

835.90 

1753.2 

67.45    68.24 

887.02 

36 

38 

1657.6 

60.26 

60.90 

837.61 

1756.5 

67.71     68.50 

888.73 

38 

40 

1660  9 

60.50 

61.14 

839.31 

1759.8 

67.96    68.76 

890.44 

40 

42 

1664.2 

60.74 

61.39 

841.01 

1703.1 

68.21     69.03 

892.15 

42 

44 

1667.5 

60.99 

61.64 

842.72 

1766  3 

68.47    69.29 

893.86 

44 

46 

1670.8 

61.23 

61.89 

&44.42 

1769.6 

68.73    69.56 

895.56 

46 

48 

1674.1 

61.48 

62.14 

846.13 

1772.9 

68.99    69.82 

897.27 

48 

50 

1677.4 

61  72 

62.39 

847.8:3 

1776  2 

69.25    70.09 

898.98 

50 

52 

1680.7 

61.96 

62.64 

849.53 

1779.5 

69.50    70.36 

900.69 

52 

54 

1684.0 

62.21 

62.89 

851  24 

1782.8 

69.76    70.62 

902.40 

54 

56 

1687.3 

62.45 

63.14 

852.94 

r786.1 

70.02    70.89 

904.10 

56 

58 

1690.6 

62.70 

63  39 

854.65 

1789.4 

70.28    51.15 

905.81 

58 

60 

1693.9 

62. 9  J 

03  64 

8.56  35 

1792.7 

70.54    71.42 

907.52 

60 

IX.— FUNCTIONS  OF  A  ONE-DE&IiEe"  CURVE.     273 


/ 

18°           1 

19°           1 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

1792.7 

70.54 

71.42 

907.52 

1S91.5 

78.58 

79.65 

958.86 

0 

1796.0 

70.80 

71.69 

909.23 

1894.8 

78.86 

79.94 

900.57 

2 

4 

1799.3 

71.06 

71.96 

910.94. 

1898.1 

79.13- 

80.22 

962.30 

4 

6 

1802.6 

71.33 

72.23 

912.65 

1901.3 

79.41 

80.51 

964.00 

6 

8 

1805.9 

71.59 

72.50 

914.36 

1904.6 

79.68 

80.79 

965.72 

8 

10 

1809.2 

71.85 

72.77 

916.07 

1907.9 

79.96 

81.08 

967.43 

10 

1-^ 

1812.5 

72.12 

73.04 

917.78 

1911.2 

80.24 

81.37  . 

969.15 

12 

14 

1815.7 

72.38 

73.31 

919.49 

1914.5 

80.51 

81.65 

970.86 

14 

16 

1819.0 

72.64 

73.58 

921.20 

1917.8 

80.79 

81.94 

972.58 

16 

18 

1822.3 

72  91 

73.85 

922.91 

1921.0 

81.07 

82.22 

974.29 

18 

20 

1825.6 

73.17 

74.12 

924.  ers 

1924.3 

81.35 

82.51 

976.01 

20 

22 

1828.9 

73.43 

74.39 

926.34 

1927.6 

81.63 

82.80 

977.72 

22 

24 

1832.2 

73.70 

74.67 

928.05 

1930.9 

81.91 

83.09 

979.44 

24 

26 

1835.5 

73.97 

74.94 

929.76 

1934.2 

82.20 

83.38 

981.15 

26 

28 

1838.8 

74.24 

75.22 

931.47 

1937.5 

82.48 

83.67 

982.86 

28 

30 

1842.1 

74.51 

75.49 

933.18 

1940.7 

82.76 

83.97 

984.58 

30 

32 

1845.4 

74.77 

75.77 

934.89 

1944.0 

83.05 

84.26 

986.30 

32 

34 

1848.7 

75.04 

76.04 

936.60 

1947.3 

83.33 

84.55 

988.02 

34 

36 

1852.0 

75.31 

76.32 

938.32 

1950.6 

83.61 

84.84 

989.74 

36 

38 

1855.3 

75.58 

76.59 

940.03 

1953.9 

83.90 

85.13 

991.46 

38 

40 

1858.6 

75.85 

76.87 

941.74 

1957.2 

84.18 

85.43 

993.18 

40 

42 

1861.9 

76.12 

77.14 

943.45 

1960.4 

84.47 

85.73 

994.90 

42 

44 

1865.1 

76.39 

77.42 

945.16 

1963.7 

84.75 

86.02 

996.62 

44 

46 

1868.4 

76.67 

77.70 

946.88 

1967.0 

85.04 

86.32 

998.34 

46 

48 

1871.7 

76.94 

77.98 

948.59 

1970.3 

85.32 

86.61 

1000.0 

48 

50 

1875.0 

77.21 

78.26 

950.30 

1973.6 

85.61 

86.91 

1001.8 

50 

52 

1878.3 

77.49 

78.53 

952.01 

1976.9 

85.90 

87.21 

1003.5 

52 

54 

1881.6 

77.76 

78.81 

953.72 

1980.1 

86.19 

87.50 

1005.2 

54 

56 

1884.9 

78.03 

79.09 

955.44 

1983.4 

86.47 

87.80 

1006.9 

56 

58 

1888.2 

78.31 

79.37 

957.15 

1986.7 

86.76 

88.09 

1008.6 

58 

60 

1891.5 

78.58 

79.65 

958  86 

1990.0 

87.05 

88.39 

1010.4 

60 



/ 

20°          1 

21°           1 

, 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

1990.0 

87.05 

88.39 

1010.4 

2088.5 

95.95 

97.58 

1062,0 

0 

o 

1993.3 

87.34 

88.69 

1012.1 

2091.8 

96.26 

97.90 

1063.7 

2 

4 

1996.6 

87.63 

88.99 

1013.8 

2095.0 

96.56 

98.21 

1065.4 

4 

6 

1999.8 

87.92 

89.29 

1015.5 

2098.3 

96.87 

98.53 

1067.2 

6 

8 

2003.1 

88.21 

89.59 

1017.2 

2101.6 

97.17 

98.84 

1068.9 

8 

10 

2006.4 

88.50 

89.89 

1019.0 

2104.9 

97.48 

99.16 

1070.6 

10 

12 

2009.7 

88.79 

90.19 

1020.7 

2108.1 

97.79 

99.48 

1072.4 

12 

14 

2013.0 

89.08 

90.49 

1022.4 

2111.4 

98.09 

99.79 

1074.1 

14 

16 

2016  3 

89.37 

90.79 

1024.1 

2114.7 

98.40 

100.1 

1075.8 

16 

18 

2019.5 

89.66 

91.09 

1025.8 

2118.0 

98.70 

J00.4 

1077.5 

18 

20 

2022.8 

89.96 

91.40 

1027.6 

2121.2 

99.00 

100.7 

1079.3 

20 

22 

2026.1 

90.25 

91.71 

1029.3 

2124.5 

99.30 

101.1 

1081.0 

22 

24 

2029-4 

90.55 

92.01 

1031.0 

2127.8 

99.60 

101.4 

1082.7 

24 

26 

2032.7 

90.85 

92.32 

1032.7 

2131.0 

99.90 

101.7 

1084.4 

26 

28 

2036.0 

91.15 

9?".  62 

1034.4 

2134.3 

100.2 

102.0 

1086.2 

28 

30 

2039.2 

91.45 

92  .«3 

1036,1 

2137.6 

100.5 

102.3 

1087.9 

30 

32 

2042.5 

91.74 

93.24 

1037.9 

2140.9 

100.8 

102.7 

1089.6 

32 

34 

2045.8 

92.04 

93.54 

1039.6 

2144.1 

101.1 

103.0 

1091.3 

34 

36 

2049.1 

92.34 

93.85 

1041.3 

2147.4 

101.4 

103.3 

1093.1 

36 

38 

2052.4 

92.64 

94.15 

10-13.0 

2150.7 

101.7 

103.6 

1094.8 

38 

40 

2055.7 

92.94 

94.46 

1044.8 

2154.0 

102.1 

104.0 

1096.5 

40 

42 

2058.9 

93.24 

94.78 

1046.5 

2157.2 

102.4 

104.3 

1098.3 

42 

44 

2062.2 

93.54 

95.09 

1048.2 

2160.5 

102.7 

104.6 

1100.0 

44 

46 

2065.5 

93.84 

95.40 

1049.9 

2163.8 

103.0 

104.9 

1101.7 

46 

48 

2068.8 

94.14 

95.71 

1051.7 

2167.1 

103.3 

105.3 

1103.4 

48 

50 

2072.1 

94.44 

96.03 

10.53.4 

2170.3 

103.6 

105.6 

1105.2 

50 

52 

2075.4 

94.74 

96.34 

1055.1 

2173.6 

103.9 

105.9 

1106.9 

52 

54 

2078.6 

95.04 

96.65 

1056.8 

2176.9 

104.2 

106.3 

1108.6 

54 

56 

2081.9 

95.34 

96.96 

1058.6 

2180.1 

104.5 

106.6 

1110.3 

56 

58 

2085.2 

95.64 

97.27 

1060. S 

2183.4 

104.8 

106.9 

1112.1 

58 

60 

2088.5 

95.95 

97.58 

1062.0 

2188  7 

105.2 

107.2 

1118.8 

60 

274      IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


/ 

22°           1 

2 

3° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

2284.8 

M. 

115.0 

E. 
117.4 

T. 

1165. S 

0 

2186.7 

105.2 

107.2 

1113.8 

0 

2 

2190.0 

105.6 

107.6 

1115.5 

22SS  1 

115.3 

117.7 

1167.5 

2 

4 

:^193.2 

105.9 

107.9 

1117.3 

2291.3 

115.7 

118.1 

1169.2 

4 

6 

2196.5 

106.2 

108.2 

1119.0 

2294.6 

116.0 

118.4 

1171.0 

6 

8 

2i99.8 

106.5 

108.6 

1120.7 

2297.8 

116.4 

118.8 

1172.7 

8 

10 

2203.0 

106.8 

10S.9 

1122.4 

2.301 . 1 

116.7 

119.1 

1174.4 

10 

VZ 

2206.3 

107,1 

109.2 

1124.2 

2.304.4 

117.0 

119.5 

1176.2 

12 

14 

2209.6 

107.4 

109.6 

1125.9 

2307.6 

117.4 

119.8 

1177.9 

14 

16 

2212  9 

107.7 

109.9 

1127.6 

2310.9 

117.7 

120.2 

1179.7 

16 

18 

2216.1 

108.0 

110.2 

1129.4 

2314  1 

118.1 

120.5 

1181.4 

18 

20 

2219.4 

108.4 

110.6 

1131.1 

2317.4 

118.4 

120.9 

1183.1 

20 

22 

2222.7 

108.7 

110.9 

1132  8 

2320.7 

118  7 

121.2 

1184.9 

22 

24 

2225.9 

109  0 

111.2 

1134.6 

2.323.9 

119.1 

121  6 

1186.6 

24 

26 

2229.2 

109.4 

111  6 

1136.3 

2327.^ 

119.4 

121.9 

1188.4 

26 

28 

2232.5 

109.7 

111.9 

11.38.0 

2330.4 

119.8 

122.3 

1190.1 

28 

30 

2235.7 

110.0 

112.3 

1139.7 

2333.7 

120.1 

122.6 

1191.8 

30 

32 

2239.0 

110.4 

112  6 

1141.5 

2337.0 

120.4 

123.0 

1193.6 

S2 

34 

2242  3 

110.7 

112.9 

1143  2 

2340.2 

120.8 

123.3 

1195.3 

34 

36 

2245.6 

111.0 

113.3 

1144.9 

2843.5 

121.1 

123.7 

1197.1 

36 

38 

2248.8 

111.4 

113.6 

1140.7 

2346.7 

121.5 

124.1 

1198.8 

38 

40 

2252.1 

111.7 

113.9 

1148.4 

2.3.50  0 

121.8 

124.4 

1200.5 

40 

42 

2255.4 

112  0 

114.3 

11.50.1 

2353  3 

122. 1 

124.8 

1202.3 

42 

44 

2258.6 

112.3 

114.6 

1151.9 

2356.5 

122  5 

125.1 

1204.0 

44 

46 

2261.9 

112.7 

115.0 

11.53.6 

2359.8 

122.8 

125.5 

1205.8 

46 

48 

2265.2 

113.0 

115.3 

11.55.4 

2.363.0 

123.2 

125.8 

1207.5 

48 

50 

2268.4 

113.3 

115.7 

11.57.1 

23(J6.3 

123.5 

126.2 

1209.2 

50 

52 

2271.7 

113.7 

iie.o 

1I.5S.8 

2369.6 

123.8 

126.6 

1211.0 

52 

54 

2275.0 

114.0 

116.3 

1100.6 

2372  8 

124.2 

126.9 

1212.7 

54 

56 

2278.3 

114.3 

116.7 

1162.3 

2376.1 

124.5 

127  3 

1214.5 

56 

58 

2281.5 

114.7 

117.0 

1104.0 

2379.3 

124.9 

127.6 

1216.2 

58 

60 

2284.8 

115.0 

117.4 

1165.8 

2382.6 

125.2 

128  0 

1218.0 

60 

/ 

24°           1 

2 

5° 

/ 

L.  r. 

M. 

E. 

T. 

L  C. 

M. 

E. 

T. 

0 

23.^2.6 

125.2 

128.0 

1218.0 

2480.4 

135.8 

139.1 

1270.3 

0 

2 

2.385.9 

125.5 

128.4 

1219.7 

2483.6 

136.2 

139.5 

1272.0 

2 

4 

23S9.1 

125.9 

128.7 

1221.4 

2486.9 

136.5 

139.9 

1273.8 

4 

6 

2.392.4 

126.2 

120.1 

1223.2 

24!0  1 

136.9 

140.3 

1275.5 

6 

8 

2395.6 

126.6 

129.5 

1224  9 

2493.4 

137.2 

140.6 

1277.3 

8 

10 

2398.9 

126  9 

129.8 

1226.7 

2496.6 

137.6 

141.0 

1279.0 

10 

12 

2402.2 

127.3 

130.2 

1228.4 

2499.9 

138  0 

141.4 

1280.8 

12 

14 

2405.4 

127  6 

130  6 

1230.2 

2.503.1 

1.38.3 

141  8 

1282.5 

14 

16 

24aS.7 

128.0 

130.9 

1231.9 

2506  4 

138.7 

142.2 

1284.3 

16 

18 

2411.9 

128.3 

131.3 

1233.6 

2509.6 

139.0 

142.5 

1286.1 

18 

20 

2415.2 

128.7 

131.7 

1235  4 

2512.9 

139.4 

142.9 

1287.8 

20 

22 

2-118  5 

129.0 

132.0 

1237.1 

2511).! 

139.8 

143.3 

1289.6 

22 

24 

2121.7 

129.4 

132.4 

123^.9 

2519.4 

140.1 

143.7 

1291.3 

24 

26 

2425.0 

129.7 

132  8 

1240.6 

2522.6 

140.5 

144.1 

1293.1 

26 

28 

242S.2 

130.1 

133  1 

1242.4 

2.525.9 

140.8 

144.5 

1294.8 

28 

30 

2431.5 

130.4 

133.5 

1244.1 

2529  1 

141.2 

144  9 

1290.6 

30 

32 

2434.8 

130  8 

133.9 

1215.8 

2.532.4 

14;  6 

145  3 

1298.3 

32 

34 

243S.0 

131.1 

134.2 

1247.6 

2535.6 

142  0 

145.6 

1300.1 

34 

36 

2441.3 

131.5 

134.6 

1J4!).3 

2.538.9 

142.3 

146.0 

1301.8 

36 

38 

2414.5 

131.8 

135.0 

1251.1 

2.542.1 

142  7 

146.4 

1303.6 

38 

40 

2447.8 

132.2 

135.4 

12.52.8 

2545.4 

143.1 

146.8 

1305.3 

40 

42 

2451.1 

132  6 

135.7 

1254.6 

2548.6 

14  5  5 

147.2 

1307.1 

42 

44 

2454.3 

132.9 

136.1 

12.56.3 

2.551.9 

143.8 

147.6 

1308.8 

44 

46 

2457.6 

133.3 

136.5 

12.58.1 

25.55.1 

144.2 

148.0 

1310.6 

46 

48 

2460.8 

133.6 

136.9 

12.59.8 

2558.4 

144.5 

148  4 

1312.4 

48 

50 

2464.1 

134.0 

137.2 

1261.5 

2561.6 

144.9 

148.8 

1314.1 

50 

52 

2467.4 

134.4 

137.6 

1263  3 

2564.9 

145.3 

149.2 

1315.9 

52 

54 

2170.6 

134.7 

138.0 

1265.0 

2508.1 

145.7 

149.5 

1317.6 

54 

56 

2473.9 

135.1 

138.4 

1266  8 

2571.4 

146.0 

149.9 

1319.4 

56 

58 

2477.1 

135.4 

138.7 

126S.5 

2.574.6 

146.4 

1.50.3 

1321.1 

58 

60 

2480.4 

135.8 

139.1 

1270.3 

2577.9 

146  8 

1.50.7 

1322.9 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGKEE  CURVE.       275 


/ 

26° 

27» 

1 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

2577.9 

146.8 

150.7 

1322.9 

2675.3 

158.3 

162.8 

1375.6 

0 

2 

2581 . 1 

147.1 

151.1 

1324.6 

2678.5 

1.58.6 

163.2 

1377.4 

2 

4 

2584.4 

147.5 

151.5 

1326.4 

2681.8 

159.0 

163.7 

1379.2 

4 

6 

2587.6 

147.9 

151.9 

1328.1 

2685.0 

1.59.4 

164.1 

1380.9 

6 

8 

2590.9 

148.3 

152.3 

1329.9 

2688.2 

159.8 

164.5 

1882.7 

8 

10 

2594.1 

148.7 

152.7 

1331.6 

2691.5 

160.2 

164.9 

1384.5 

10 

12 

2597.4 

149.1 

l.=)3.l 

1333.4 

2694.7 

160.6 

165.3 

1386.2 

12 

14 

2600.6 

149.4 

153.5 

1335.2 

2698.0 

181  0 

165.7 

i:J88.0 

14 

10 

2603.9 

149.8 

153.9 

13.36.9 

2701.2 

161.4 

166  1 

1389.8 

16 

18 

2607.1 

150  2 

154.3 

1338.7 

2704.4 

161  8 

166.5 

1391.5 

18 

20 

2610.4 

150  6 

154.7 

1340.4 

2707.7 

162.2 

167.0 

1393.3 

20 

>_)•> 

2613.6 

151  0 

155.1 

1342.2 

2710.9 

162.6 

167.4 

1395.0 

22 

24 

2616.9 

151.4 

155.5 

1343.9 

2714.1 

163  0 

167.8 

1396.8 

24 

26 

2620.1 

151.7 

155.9 

1945.7 

2717.4 

163.4 

168.2 

1398.6 

26 

28 

2623.4 

152.1 

156.3 

1347.4 

2720.6 

163.8 

168.6 

1400.3 

28 

30  • 

2626.6 

152.5 

156.7 

1349.2 

2723.8 

164.2 

169.1 

1402.1 

30 

32 

2629.8 

152.9 

157.1 

1351.0 

2727.1 

164.6 

169.5 

1403.9 

32 

34 

2633.1 

153.3 

157.5 

1352.7 

2730.3 

165.0 

169.9 

1405.6 

34 

36 

2636  3 

153.7 

157.9 

1354  5 

2733.0 

165.4 

170.3 

1407.4 

36 

38 

2639.6 

154.0 

158.3 

1356.2 

2736.8 

165.8 

170.8 

1409.2 

38 

40 

2642.8 

154  4 

158.7 

1358.0 

2740.0 

166.2 

171.2 

1410.9 

40 

42 

2610.1 

154.8 

159.1 

1359.8 

2743  3 

166.6 

171.6 

1412.7 

42 

44 

2649.3 

155.2 

159.5 

1301.5 

2746.5 

167.0 

172.0 

1414.5 

44 

46 

2652.6 

155.6 

160.0 

1.363.3 

2749  7 

167  4 

172.5 

1416.3 

46 

48 

2655.8 

156.0 

160.4 

1365.1 

2753.0 

167.8 

172.9 

1418.0 

48 

50 

2659.1 

150.3 

160.8 

1366.8 

2756.2 

168.2 

173  3 

1419.8 

50 

52 

2662.3 

156.7 

161.2 

1368.6 
13-0.4 

2759.5 

168.6 

173.7 

1421.6 

52 

54 

2665. G 

157.1 

161.6 

2762.7 

169.0 

174.1 

1423.3 

54 

56 

2668.8 

1.57  5 

162.0 

13~2.1 

2765.9 

169.4 

174.6 

1425.1 

56 

58 

2672.1 

157.9 

162.4 

1373.9 

2769.2 

169.8 

175.0 

1426.9 

58 

60 

2675.3 

1.58.3 

162.8 

1375.6 

2772.4 

170.2 

175.4 

1428.6 

60 

0 

2S 

i° 

'2S 

1" 

/ 

L.  C. 

M. 

E. 

T. 

L.C. 

M. 

E. 

T. 

2772.4 

170.2 

175  4 

1428.6 

2869.4 

182.5 

188.5 

1481.9 

0 

2 

2775.6 

170.6 

175.8 

1430  4 

2872.6 

182.9 

189.0 

1483.7 

2 

4 

2778.9 

171.0 

176.3 

1432.2 

2875.8 

183  3 

189.4 

1485.4 

4 

6 

2782.1 

171.4 

176  7 

1434.0 

2879.1 

183.7 

189.9 

1487.2 

6 

8 

278.J.3 

171.8 

177  1 

1435.7 

2882.3 

184.2 

190.3 

1489.0 

8 

10 

2788.6 

172.2 

177.6 

1437.5 

2885.5 

184.6 

190.8 

1490.8 

10 

12 

2791 .8 

172.6 

178.0 

1439.3 

2888.7 

185.0 

191.2 

1492.6 

12 

14 

2795.0 

173  0 

178.4 

1441.1 

2892.0 

185.4 

191.7 

1494.3 

14 

16 

2798.3 

173.4 

17S.9 

1442  8 

2895.2 

185.8 

192.1 

1496.1 

16 

18 

2801.5 

173.8 

179.3 

1444.6 

2898.4 

186.3 

192.5 

1497.9 

18 

20 

2804.7 

174.3 

179.7 

1446.4 

2901.6 

186.7 

193.0 

1499.7 

20 

22 

2808.0 

174.7 

180.2 

1448.2 

2904.8 

187.1 

193.5 

1501.5 

22 

24 

2811.2 

175.1 

180.6 

1449.9 

2908.1 

187.5 

193.9 

1503.2 

24 

26 

2814.4 

175.5 

181.0 

1451.7 

2911.3 

188.0 

194.4 

1505.0 

26 

28 

2817.7 

175.9 

181  5 

1453.5 

2914  5 

188.4 

194.8 

1506.8 

28 

30 

2820.9 

176  3 

181.9 

1455.2 

2917.7 

188.8 

195.3 

1508-6 

30 

32 

2824.1 

176.7 

182.3 

14.57.0 

2921.0 

189.2 

195.7 

1510.4 

32 

34 

2827  4 

177.1 

182.8 

14.58.8 

2924.2 

189.7 

196.2 

1.512.1 

34 

36 

28.30.6 

177.5 

183.2 

1460.6 

2927.4 

190.1 

196.7 

1513.9 

36 

38 

2833.8 

177.9 

183.6 

1462  3 

2930.6 

190.5 

197.1 

1515.7 

38 

40 

2837.1 

178.4 

184.1 

1464.1 

2933.9 

190.9 

197.6 

1517.5 

40 

42 

2840  3 

178.8 

184.5 

1465.9 

2937.1 

191.4 

198.0 

1519.3 

42 

44 

2843  5 

179  2 

185.0 

1467.7 

2940.3 

191.9 

198.5 

1521.0 

44 

46 

2846.8 

179.6 

185.4 

1469,5 

2943.5 

192.4 

198.9 

1522.8 

46 

48 

2850.0 

180  0 

185.9 

1471.2 

2946.8 

192.8 

199.4 

1524.6 

48 

50 

2853.2 

180.4 

186.3 

1473.0 

2950  0 

193.2 

199.8 

1526.4 

50 

52 

2856  5 

180.8 

186.8 

1474.8 

2953.2 

193.6 

200.3 

1528.2 

52 

54 

2859.7 

181.2 

187.2 

1476.6 

2956.4 

194.0 

200.8 

1.530.0 

54 

56 

2862.9 

181.6 

187.6 

1478.3 

29.59.6 

194  4 

201.2 

1531.7 

56 

58 

2866.2 

182  0 

188.1 

1480.1 

2962.9 

194.8 

201.7 

1.5.33.5 

58 

60 

2809.4 

182  5 

188.5 

1481  9 

2966.1 

195.2 

202.1 

1.535  3 

60 

276 

IX.     FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 

1 

30° 

L.  C. 

3 

M. 

1° 

1 

L.  C. 

M. 

E. 

T. 

E. 

T. 

0 

2966.1 

195.2 

202.1 

1535.3 

3062.6 

208.4 

216  3 

1589.0 

0 

2 

2969.3 

195.6 

202.6 

1537.1 

3065.8 

208.8 

216.8 

1590.8 

2 

4 

2972.5 

196.1 

203.1 

1538.9 

3069.0 

209.3 

217.2 

1592.6 

4 

6 

2975.7 

196.5 

203.5 

1540.7 

3072.2 

209.7 

217.7 

1594.4 

6 

8 

2979.0 

197.0 

204.0 

1542.5 

3075.4 

210.2 

218  2 

1596.2 

8 

10 

2982.2 

197  4 

204.5 

1544  3 

3078.6 

210.6 

218.7 

1598.0 

10 

12 

2985.4 

197.8 

204.9 

1546.0 

3081.8 

211.1 

219.2 

1599.8 

12 

14 

2988.6 

198.2 

205.4 

1547.8 

3085.0 

211.5 

219.6 

1601.6 

14 

16 

2991.8 

198.6 

205  9 

1549.6 

3088.3 

212.0 

220  1 

1603.4 

16 

18 

2995  0 

199.1 

206  3 

1.551.4 

3091.5 

212.4 

220.6 

1605.2 

18 

20 

2998.3 

199.5 

206.8 

1553.2 

3094.7 

212.9 

221.1 

1607.0 

20 

22 

3001.5 

199.9 

207.3 

1555.0 

3097.9 

213.3 

221.6 

1608.8 

22 

24 

3004.7 

200.4 

207.7 

1.556.8 

3101.1 

213.8 

222.1 

1610.6 

24 

26 

3007.9 

200.8 

208.2 

1558.6 

3104.3 

214.2 

222.6 

1612.4 

26 

28 

3011.1 

201.3 

208.7 

1560.4 

3107.5 

214.7 

223.0 

1614.2 

28 

30 

3014.3 

201.7 

209.1 

1562.2 

3110.7 

215.1 

223.5 

1616.0 

30 

32 

3017.6 

202  1 

209.6 

1564.0 

3113.9 

215.6 

224.0 

1617.8 

32 

34 

3020.8 

202  6 

210.1 

1 565 . 7 

3117.1 

216.0 

224.5 

1619.6 

34 

36 

3024.0 

203.0 

210.5 

1567.5 

3120.3 

216.5 

225.0 

1621.4 

36 

38 

3027.2 

203.5 

211.0 

1569.3 

3123.5 

216.9 

225.5 

1623.2 

38 

40 

3030  4 

203.9 

211  5 

1571.1 

3126.7 

217.4 

226.0 

1625.0 

40 

42 

3033.6 

204.3 

212  0 

1.572.9 

3129.9 

217. S 

226.5 

1626.8 

42 
44    \ 

44 

3036.9 

204.8 

212  4 

1.574.7 

3133.1 

218.3 

227.0 

1628.6 

46 

3040.1 

205.2 

212.9 

1576.5 

3136.4 

218.7 

227.5 

16-30.5 

46     ' 

48 

3043.3 

205.7 

213.4 

1578.3 

3139.6 

219.2 

228.0 

16:32.3 

48 

50 

3046.5 

206.1 

213.9 

1580.1 

3142.8 

219.6 

228.4 

16:34.1 

50 

52 

3049.7 

206.5 

214.4 

1581.9 

3146.0 

220.1 

228.9 

1635.9 

52 

54 

3052.9 

207  0 

214  8 

1583.7 

3149.2 

220.5 

229.4 

1637.7 

54 

56 

3056.2 

207.4 

215.3 

1585.5 

31.52.4 

221.0 

2:.'9.9 

16:39.5 

56 

58 

3059.4 

207.9 

215  8 

1587.2 

315.5.6 

221.5 

230.4 

1641.3 

58 

60 

3062.6 

208.4 

216  3 

1.589.0 

31.58.8 

222.0 

230.9 

1643.1 

60 

/ 

32°                           1 

33" 

'» 

/ 

L.  C. 

3158.8 

M. 
222.0 

E. 
2.30.9 

T. 

1643.1 

L.  C 

M. 

E. 

T. 

0 

3254  9 

2.36.0 

246.1 

1697  3 

0 

>■> 

3162.0 

222.5 

231.4 

1644  9 

3258  1 

236.4 

246.6 

1699.1 

2 

4 

3165.2 

222.9 

231.9 

1646  7 

3261.3 

2.36  9 

247.1 

1700  9 

4 

6 

3168.4 

223.4 

232.4 

1648.5 

.3264.5 

2.37  4 

247.7 

1702.7 

6 

8 

3171.6 

223.8 

2.32.9 

1650  3 

3267  7 

237  9 

248.2 

1704.5 

8 

10 

3174.8 

224.3 

233.4 

1652  1 

.3270.8 

2:38.4 

248.7 

1706.4 

10 

12 

3178.0 

224  8 

233.9 

1653.9 

.3274  0 

2:38  9 

249.2 

1708.2 

12 

14 

3181.2 

225.2 

234.4 

16.55  7 

3277  2 

2:39  3 

249.7 

1710.0 

14 

16 

3184.4 

225  7 

234.9 

1657  5 

3280  4 

2:39.8 

250.2 

1711.8 

16 

18 

3187.6 

226.1 

2:35.4 

1659.3 

328:3.6 

240.3 

250.8 

1713.6 

18 

20 

3190.8 

226.6 

235.9 

1661  1 

.3286.8 

240.8 

251.3 

1715.5 

20 

22 

3194.0 

227.1 

236.4 

1662.9 

3290  0 

241.2 

251  8 

1717.3 

22 

24 

3197  2 

227.5 

236  9 

1664.7 

3203.2 

241  7 

252  3 

1719.1 

24 

26 

3200.4 

228.0 

237.4 

1666  5 

.3296.4 

242.2 

252.9 

1720.9 

26 

28 

3203.6 

228.4 

237  9 

1668  3 

.3299.6 

242.7 

253  4 

1722.7 

28 

30 

3206.8 

228  9 

238.4 

1670.1 

3302.7 

243  2 

253.9 

1724.6 

30 

32 

3210.0 

229  4 

239.0 

1671.9 

3-305  9 

243.6 

254.4 

1726.4 

32 

34 

3213.2 

229  8 

239.5 

1673  7 

3.309  1 

244.1 

255  0 

1728.2 

34 

36 

3216  5 

230.3 

240.0 

1675  5 

.3312  3 

244.6 

255.5 

1730.0 

36 

38 

3219.7 

230.7 

240.5 

1677.4 

3315  5 

245.1 

256.0 

1731.8 

38 

40 

3222  9 

231  2 

241.0 

1679.2 

3:318.7 

245.6 

256  5 

1733.6 

40 

42 

3226  1 

231  7 

241  5 

1681.0 

3.321.9 

246  0 

257  1 

1735.5 

42 

44 

3629.3 

232.2 

242  0 

1682  8 

3325.1 

246.5 

257.6 

1737.3 

44 

46 

3232.5 

232.6 

242.5 

1684.6 

3:328.3 

247.0 

258.1 

1739.1 

46 

48 

3235.7 

233.1 

243.0 

1686  4 

.3.3:31.5 

247  5 

258  6 

1740.9 

48 

50 

3238.9 

233.5 

243.5 

1688  2 

:3:3:34.6 

248.0 

259.2 

1742.7 

50 

52 

3242.1 

234.0 

244  1 

1690.0 

33:37.8 

248.4 

259.7 

1744.6 

52 

54 

3245  3 

234.5 

244  6 

1691.8 

3341 .0 

248  9 

260  2 

1746.4 

54 

56 

3248.5 

235  0 

245.1 

1693  7 

:3.344.2 

249.4 

260.8 

1748.2 

56 

58 

3251 . 7 

235.5 

245.6 

1695  5 

:3:347.4 

249.9 

261.3 

1750.0 

58 

60 

3254  9 

236  0 

246  1 

1697  3 

:3:3.j0.6 

250.4 

261  8 

1751.8 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.  ^^1 


/ 

S4^ 

3 

0° 

/ 

L.  0. 

M. 

E. 

T. 

L.C. 
3146.1 

M. 
265.2 

E. 

278.1 

T. 

1806.7 

0 

3350.6 

250.4 

261.8 

175I.S 

0 

2 

3353.8 

2.50.8 

262.3 

1753.7 

3119.3 

265.7 

278.6 

180S.5 

0 

4 

3357.0 

251.2 

262.9 

V>'M.n 

3452.5 

266.2 

279.2 

1810.3 

4 

6 

3360.1 

251.7 

263.4 

1757.3 

3455.6 

266.7 

279.7 

1812.2 

6 

8 

3363.3 

252.2 

2o4.0 

1759.1 

3458.8 

267.2 

280.3 

1814.0 

8 

10 

3366.5 

252.7 

264.5 

1761.0 

3162.0 

267.7 

280.8 

1815.8 

10 

\2 

3369.7 

253.2 

265.0 

1762.8 

3165.2 

268.2 

281.4 

1817.7 

12 

14 

3372.9 

253.7 

265.0 

1764.6 

3468.3 

268  7 

2S1.9 

1819.5 

14 

16 

3376.1 

2.54.2 

^'66.1 

1 766 . 4 

3171.5 

269.2 

2S2 . 5 

1821.3 

16 

18 

3379.2- 

254.7 

266.7 

1768.3 

3474.7 

269.7 

2S3.0 

1823.2 

18 

20 

33S2.4 

255.2 

267.2 

1770.1 

34. r. 9 

270.2 

283.6 

1825.0 

20 

22 

3385.6 

255.7 

267.7 

1771  9 

3481.0 

270.7 

284.2 

1826.8 

22 

24 

3388.8 

256.2 

268.3 

1773.7 

3484.2 

271.2 

284.7 

1828.7 

24 

26 

3392.0 

256.7 

268.8 

1775.6 

3487.4 

271.7 

285.3 

1830.5 

20 

28 

3395.2 

257.2 

269.3 

1777.4 

3490.6 

272.2 

285.9 

1832.3 

28 

30 

3398.3 

257.7 

269.9 

1779.2 

3493.7 

272.7 

286.4 

1834.2 

30 

3i 

3401.5 

258.2 

270.4 

1781.0 

3496.9 

273.2 

287.0 

1836.0 

32 

34 

3404.7 

258.7 

271.0 

1782.9 

3500.1 

273.  V 

287.5 

1837.8 

34 

36 

3407.9 

2.59.2 

271.5 

1784.7 

3503.3 

274.2 

288.1 

1839.7 

30 

38 

3411.1 

259.7 

272  0 

1786.5 

3506.5 

274.7 

288.7 

1841.5 

38 

40 

3414.3 

260.2 

272.6 

1788.4 

3509.6 

275.2 

289.2 

1843.4 

40 

42 

3417.4 

260.7 

273.1 

1790.2 

3512.8 

275.7 

289.8 

1845.2 

42 

44 

3420.6 

261.2 

273.7 

1792.0 

3516.0 

276  2 

290.4 

1847.1 

44 

46 

3423.8 

261.7 

274.2 

1793  9 

3519.2 

^<0.   i 

290.9 

1848.9 

46 

48 

3427.0 

262.2 

274.8 

1795.7 

3522.3 

291.5 

1850.7 

48 

50 

3430.2 

262.7 

275.3 

1797.5 

3525.5 

277.7 

292.0 

1852.6 

50 

52 

3433.4 

263.2 

275.9 

1799.3 

3528.7 

278.2 

292.6 

1854.4 

52 

54 

3436.5 

263.7 

276.4 

1801.2 

3531.9 

278.7 

293.2 

1850.3 

54 

56 

3439.7 

264.2 

277.0 

1803.0 

3.5.35.0 

279.2 

293.7 

1858.1 

56 

58 

3442.9 

264.7 

277.5 

1804.8 

3538.2 

279.8 

294.3 

18.59.9 

58 

GO 

3446.1 

265.2 

278.1 

1806.7 

3541.4 

280.4 

294.9 

1861.8 

60 

/ 

3«°                          1 

3- 

?" 

/ 

L.  C. 

M. 

E. 

T. 

L.C. 

M. 

E. 

T. 

0 

3541.4 

280.4 

294.9 

1861.8 

3636.3 

296.1 

312.3 

1917.3 

0 

o 

3544.6 

280  9 

295.4 

1863.6 

3639.5 

296.6 

312.8 

1919.1 

0 

4 

3547.7 

281.4 

296.0 

1865.5 

3642  6 

297  1 

313.4 

1921.0 

4 

6 

3550.9 

281.9 

296.6 

1867.3 

3645.8 

297.7 

314  0 

1922.8 

6 

8 

3554.0 

282.5 

297.2 

1869.2 

3648.9 

298.2 

314.6 

1924.7 

8 

10 

3557.2 

283.0 

297.7 

1871.0 

3652.1 

298.7 

315.2 

1926.5 

10 

12 

3560.4 

283.5 

298.3 

1872.9 

3655.2 

299.3 

315.8 

1928.4 

12 

14 

3563.5 

284.0 

298.9 

1874.7 

36.58.4 

299.8 

316.4 

1930.2 

14 

16 

3566.7 

284.6 

299.5 

1876.5 

3661.6 

300.3 

317.0 

1932.1 

16 

18 

3569.9 

285.1 

300.0 

1878.4 

3664.7 

300.9 

317.5 

1933.9 

18 

20 

3573.0 

285.6 

300.6 

1880.2 

3667.9 

301.4 

318.1 

1935.8 

20 

22 

3576.2 

286.1 

301.2 

1882.1 

3671.0 

301.9 

318.7 

1937.6 

22 

24 

3579.4 

286.7 

301 .8 

1883.9 

3674.2 

302.5 

319.3 

1939.5 

24 

26 

3582.5 

287.2 

302:3 

18S5.8 

3677.3 

303.0 

319.9 

1941.3 

26 

28 

a585.7 

287.7 

302.9 

1887.6 

3680.5 

303.5 

320.5 

1943.2 

28 

30 

3.588.8 

288.2 

303.5 

1889.5 

3683.6 

304.1 

321.1 

1945.0 

30 

32 

3.592.0 

288.8 

304.1 

1891.3 

3686.8 

304.6 

321.7 

1946.9 

32 

34 

3595.2 

289.  S 

304.6 

1893.2 

3690.0 

305.1 

322.3 

1948.8 

34 

36 

3598.3 

289.8 

305.2 

1895.0 

3693.1 

305.7 

322.9 

1950.6 

36 

38 

3601.5 

290.3 

305.8 

1896.9 

3696.3 

306.2 

323.5 

1952.5 

38 

40 

3604.7 

290.9 

306.4 

1898.7 

3699.4 

306.7 

324.2 

1954.4 

40 

42 

3607.8 

291.4 

307.0 

1900.6 

3702.6 

307.3 

324.8 

19.56.2 

42 

44 

3611.0 

291.9 

307.5 

1902.4 

3705.7 

307.8 

325.4 

1958.1 

44 

46 

3614.1 

292.4 

308.1 

1904.3 

3708.9 

308.3 

326.0 

1960.0 

46 

48 

3617.3 

893.0 

308.7 

1906.1 

3712.1 

308.9 

326  6 

1961.8 

48 

50 

3620.5 

293.5 

309.3 

1908.0 

3715.2 

309.4 

327  2 

1963.V 

50 

52 

3623  6 

294.0 

309.9 

1909.8 

3718.4 

309  9 

327.8 

1965.5 

52 

54 

3626.8 

294.5 

310.5 

1911.7 

3721  5 

310.5 

328.4 

1967.4 

54 

56 

3630.0 

295.1 

311.1 

1913.5 

3724.7 

311.0 

329.0 

1969.3 

56 

58 

3633.1 

295.6 

311.7 

1915.4 

3727.8 

311.6 

329  6 

1971   1 

58 

60 

3636.3 

296.1 

312.3 

1917.3 

3731.0 

312.2 

330  2 

1973.0 

60   ^ 

278 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 

/ 

38°                         1 

39° 

t 

L.  C.         M. 

E. 

T. 

L.  C 
3825  5 

M 
328-7 

E. 

348.7 

T. 
2029.1 

0 

3731.0    312.2 

330.2 

1973.0 

0 

2 

3734.1     312  7 

330.8 

1974.9 

3828.6 

329.2 

349  3 

2031.0 

2 

4 

3737.3    313  3 

:i31.4 

1976-7 

3831.8 

329.8 

.349.9 

2032  9 

4 

6 

3740  4    313.8 

332.0 

1978.6 

3834  9 

3^0-3 

.350.8 

20.34.7 

6 

8 

3743.6    314.4 

332.6 

1980.5 

3838.0 

330.9 

351.2 

2036.6 

8 

10 

3746.7    314.9 

333.2 

1982.3 

3841.2 

331  5 

351.8 

2038.5 

10 

12 

3749  9    315.5 

::!33.8 

1984-2 

3844  3 

332.0 

352.4 

2040  4 

12 

14 

3753  0    316  0 

.334.5 

1986.1 

3847.4 

332.6 

353.1 

2042  3 

14 

16 

3756  2    316.6 

335.1 

1987.9 

3850.6 

333.2 

353.7 

2044.1 

16 

18 

3759.3    317.1 

335-7 

1989.8 

38.53  7 

333  7 

354-3' 

2046.0 

18 

20 

3762  5    317  7 

336  3 

1991.7 

38.56.8 

334.3 

354.9 

2047.9 

20 

22 

3765.6  .318  2 

336  9 

1993-6 

3860  0 

334.9 

355.6 

2049.8 

22 

24 

3768.8    318.8 

337.5 

1995.4 

3863.1 

;3;35.4 

3.56.2 

2051  7 

24 

26 

3771.9    319.3 

338.1 

1997.3 

:^66.2 

336. 0 

356.9 

2053.5 

26 

28 

3775  1     319.9 

338.7 

1999.2 

3869.4 

a36.6 

357  5 

2055.4 

28 

30 

3778.2    320.4 

339.4 

2001.0 

:S872.5 

337.1 

358.1 

2057-3 

30 

32 

3781.4    321.0 

310.0 

2002.9 

3875  6 

337  7 

3.58.8 

2059.2 

32 

34 

3784  5     321.5 

310-6 

2004.8 

3878.8 

338  3 

359  4 

2061.1 

34 

36 

3787.7    322.1 

341.2 

2006.6 

3881.9 

3.38  8 

.360.1 

2063.0 

36 

38 

3790  8    322.6 

341.8 

2008.5 

3885.0 

339  4 

360.7 

2064.8 

38 

40 

3794.0    323  2 

342.4 

2010  4 

3888.2 

340.0 

361.3 

2066.7 

40 

42 

3797  1     323.7 

343.1 

2012  3 

3891.3 

:i40.5 

362.0 

2068.6 

42 

44 

3800.3    324.3 

:^3  7 

2014.1 

3894  4 

341  1 

362.6 

2070.5 

44 

46 

3803.4    324.8 

344.3 

2016  0 

3897.6 

341.7 

363.3 

2072.4 

46 

48 

3806.6    325.4 

344.9 

2017.9 

3900  7 

342  2 

363.9 

2074  2 

48 

50 

3809.7    325  9 

345.6 

2019.7 

3903.8 

342.8 

364.5 

2076  1 

50 

52 

3812.9    326  5 

M6  2 

2021.6 

3907  0 

343  4 

365.2 

2078.0 

52 

54 

3816  0    327  0 

346.8 

2023.5 

3910.1 

343.9 

365.8 

2079.9 

M 

56 

3819.2    327.6 

347  4 

2025  4 

3913  2 

.344.5 

366.5 

2081.8 

56 

68 

3822.3    328.1 

348  1 

2027.2 

3916.4 

345  1 

367  1 

2083.7 

58 

60 

3825.5    328.7 

348.7 

2029.1 

3919  5 

U5.6 

367  7 

2085.5 

60 

/ 

40'^                          1 

41° 

' 

L.  C.       M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

3919.5    345.6 

367.7 

2085.5 

4013.4 

362.9 

387.4 

2142.3 

0 

2 

392J.6    346.1 

368.4 

2087  4 

4016.5 

363.4 

388.1 

2144.2 

2 

4 

3925.8    346  7 

369  0 

2U89.3 

4019.6 

364.0 

388.8 

2146.1 

4 

6 

3928  9    347  2 

369.7 

2091  2 

4022.7 

364  5 

389.4 

2148.0 

6 

8 

3932  0    347  8 

370.3 

2093. 1 

4025.9 

365.1 

390  I 

2149.9 

8 

10 

3935.1     348.4 

371  0 

2095  0 

4029.0 

365.6 

390.7 

21.51.9 

10 

12 

3938.3    a48.9 

371  0 

2096  9 

4032.1 

366.2 

391.4 

2153.8 

12 

14 

3941.4    349.5 

372.3 

2098.8 

40-35.2 

366.8 

392.1 

2155.7 

14 

16 

3944.5     350.1 

372.9 

2100.7 

4038.3 

367.4 

392.7 

2157.6 

16 

18 

3947.7    350.7 

373.6 

2102  6 

4041.4 

368.0 

393.4 

2159.5 

18 

20 

3950.8    351.3 

374.3 

2104.5 

4044.6 

368.6 

394.1 

2161.4 

20 

22 

39.53.9    351  8 

374.9 

2106.3 

4047.7 

369.2 

394.7 

2163.3 

22 

24 

3957.1     352.4 

375.6 

2108.2 

4050.8 

.369.8 

395.4 

2165.2 

24 

26 

3960.2    353.0 

376.2 

2110.1 

4053  9 

370.4 

.396.1 

2167.1 

26 

28 

3963.3    3.53  6 

376.9 

2112  0 

40.">7.0 

371.0 

396.8 

2169.0 

28 

30 

3966.4    354.2 

377.5 

2113  9 

4060.1 

371.6 

.397.5 

2170  9 

30 

32 

3969.6    354.7 

378.2 

2115  8 

4063.3 

372.2 

398.1 

2172.8 

32 

34 

3972.7    355.3 

378.8 

2117  7 

4066.4 

372.8 

.398.8 

2174.7 

34 

36 

3975.8    355.9 

379.5 

2119  6 

4069.5 

373.4 

399.5 

2176.6 

36 

38 

3979.0    356.5 

380  1 

2121.5 

4072.6 

374.0 

400.2 

2178.5 

38 

40 

3982.1     357.1 

380.8 

2123.4 

4075.7 

.374.6 

400.9 

2180.4 

40 

42 

3985.2    357  6 

381  4 

2125  3 

4078. 8 

375.2 

401.5 

2182.4 

42 

44 

3988.4    358.2 

382.1 

2127.2 

4082.0 

375.8 

402.2 

2184.3 

44 

46 

3991  5    358  8 

382  8 

2129.1 

4085.1 

376.4 

402.9 

2186.2 

46 

48 

3994.6    359.4 

383.4 

2131.0 

408«.2 

377.0 

403.6 

2188.1 

48 

50 

3997  7    360  0 

384.1 

2132.9 

4091.3 

377.6 

404.3 

2190.0 

50 

52 

4000  9    :^60  5 

384  8 

2134.7 

4094.4 

378.2 

404.9 

2191.9 

52 

54 

4004  0    361  1 

385.4 

2136.6 

4097.5 

.378.8 

405.6 

2193.8     ! 

54 

56 

4007  1     .361.7 

386.1 

2138.5 

4100.7 

379.4 

400.3 

2195.7 

56 

58 

4010  3     3ti2.3 

380  8 

2140  4 

4103.8 

380  0 

407.0 

2197.6 

58 
60 

CO 

4013  4    3'i2.9 

3!S7  4 

2142  3 

4106.9 

380.6 

407.7 

2199.5     ■ 

IX.-FUNCTIONS  OF  A   ONE-DEGREE  CURVE.     279 


r 

4 

2" 

43° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

/ 

0 

4106.9 

380.6 

407.7 

2199.5 

4200.1 

398.7 

428.6 

2257.1 

0 

2 

4110.0 

381.2 

408.3 

2201.4 

4 -,'03. 2 

399.3 

429.3 

2259.0 

2 

4 

4113.1 

381.8 

409.0 

2203.3 

4206.3 

399.9 

430.0 

2261 .0 

4 

6 

4116.2 

382.4 

409.7 

2205.3 

4209.4 

400.5 

430.7 

2262.9 

6 

8 

4119.3 

383.0 

410.4 

2207.2 

4212.5 

401.1 

431.4 

2264.8 

8 

10 

4122.4 

383.6 

411.1 

2209.1 

4215.6 

401.7 

432.1 

2260.7 

10 

13 

4125.5 

3S4 . 2 

411.8 

2211.0 

4218.7 

402.4 

432.8 

8268.7 

12 

14 

4128.6 

384.8 

412.5 

2212.9 

4221.8 

403.0 

433.5 

2270.6 

14 

16 

4131.8 

38.  4 

413.2 

2214.9 

4224.9 

403.6 

434.2 

2272.5 

16 

18 

4134.9 

386  0 

413.9 

2216.8 

4228.0 

404.2 

434,9 

2274.5 

18 

20 

4138.0 

380  i 

414.6 

2218.7 

4231.1 

404.8 

435.6 

2276.4 

20 

22 

4141.1 

3S7  2 

415.3 

2220.6 

4234.2 

405 . 4 

436.3 

2278.3 

22 

24 

4144.2 

387.8 

416.0 

2222.5 

4237.3 

406.1 

437.0 

2280.2 

24 

26 

4147  3 

388.4 

416.6 

2224.4 

4240.4 

406  7 

437  8 

2282.2 

26 

2S 

4150.4 

389  0 

417.3 

2.226.4 

4243.5 

407.3 

438.5 

2281.1 

28 

30 

4153.5 

389.6 

418.0 

2228.3 

4246.5 

407.9 

439  2 

2286.0 

30 

32 

4156.6 

390,2 

418.7 

2230.2 

4249.6 

408  5 

439.9 

2288.0 

32 

34 

4159.7 

390.8 

419.4 

2232.1 

4252.7 

409.1 

440.6 

2289.9 

34 

36 

4162  8 

391.4 

420.1 

2234.0 

4255.8 

409  8 

441.4 

2291.8 

36 

38 

4165.9 

392.0 

420.8 

2-J36.0 

4258.9 

410.4 

442.1 

2293.8 

38 

40 

4169.0 

392.6 

421.5 

2237.9 

4262.0 

411.0 

442.8 

2295.7 

40 

42 

4172.1 

393.2 

422  2 

2239.8 

4205.1 

411.6 

443.5 

2297.7 

42 

44 

4175.2 

393.8 

422.9 

2241.7 

4208.2 

412  2 

444.2 

2299.6 

44 

46 

4178.4 

394.4 

423.6 

2243.6 

4271.3 

412.8 

445.0 

2.301.5 

46 

48 

4181.5 

395.0 

424.3 

2245.6 

4274.4 

413.5 

445.7 

2303.5 

48 

50 

4184.6 

395.6 

425.0 

2247.5 

4277.5 

414.1 

446.4 

2305.4 

50 

52 

4187.7 

396.2 

425.7 

2i49  4 

4280.6 

414.7 

447.1 

2307.3 

52 

54 

4190.8 

396.8 

426.4 

2251.3 

4283.7 

415  3 

447.8 

2309.3 

54 

56 

4193.9 

397  4 

427.1 

2--"53.3 

42,S6.8 

415.9 

448,6 

2311.2 

56 

58 

4197.0 

398  0 

427.8 

2255  2 

4289.9 

416.5 

449.3 

2313.1 

58 

60 

4-'00.1 

398.7 

428.6 

22.-.7.1 

4293.0 

417.2 

450.0 

2315.1 

60 

/ 

44°           1 

4 

5°   » 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

4385.5 

M. 

436.2 

E. 
472.1 

T. 
2373.4 

0 

4293.0 

417.2 

450.0 

2315.1 

0 

2 

4296  1 

417.8 

450.7 

2317.0 

4388.6 

436.8 

472.9 

2375.4 

2 

4 

4299.2 

418.4 

451.5 

2319.0 

4391.7 

437.5 

473.6 

2377.3 

4 

6 

4302.2 

419.1 

452.2 

2320.9 

4394.7 

438.1 

474.4 

2379.3 

6 

8 

4305.3 

419.7 

452  9 

2322.8 

4397.8 

438  8 

475.1 

2381.2 

8 

10 

4308.4 

420.3 

453.7 

2324.8 

4400.9 

439.4 

475.9 

2383.2 

10 

12 

4311.5 

421.0 

454.4 

2326.7 

4404.0 

440.0 

476  6 

2385.2 

12 

14 

4314.6 

421 . 6 

455.1 

2328.7 

4407.0 

440.7 

477.4 

2387.1 

14 

16 

4317.7 

422.2 

455.9 

2330.6 

4410.1 

441.3 

478.1 

2389.1 

16 

18 

4320.7 

422.9 

456.6 

2332.6 

4413.2 

442.0 

478  9 

2391 .0 

18 

20 

4323.8 

423  5 

457.3 

2334.5 

4416.3 

442,6 

479.6 

2393.0 

20 

22 

4326.9 

424.1 

458.1 

2336.4 

4419.3 

443.2 

480.4 

2394.9 

22 

24 

4330.0 

424.8 

458.8 

2338.4 

4422.4 

443.9 

481.1 

2396.9 

26 

4333.1 

425  4 

459.5 

2340  3 

4425 . 5 

444.5 

481  9 

2398.8 

26 

28 

4336.2 

426.0 

460.3 

2342.3 

4428.6 

445.2 

482.6 

2400.8 

28 

30 

4339.2 

426.7 

461  0 

2344.2 

4431.6 

445.8 

483.4 

2402.8 

30 

32 

4342.3 

427  3 

461.7 

2346.1 

4431.7 

446  4 

484.2 

2404.7 

32 

34 

4345.4 

427  9 

462.5 

2348.1 

4437 . 8 

447.1 

484.9 

2406.7 

34 

36 

4348.5 

428  6 

463.2 

2350  0 

4440.9 

447.7 

485.7 

2408.6 

36 

38 

4351.6 

429.2 

463.9 

2352.0 

4444.0 

448  3 

486.5 

2410  6 

38 

40 

4354.7 

429.8 

464.1 

2353.9 

4447.0 

448.9 

487.2 

2412.6 

40 

42 

4357.7 

430.5 

465.4 

2355,9 

4450.1 

449  5 

488.0 

2414.5 

42 

44 

4360.8 

431.1 

466.2 

23.57.8 

4453  2 

450.2 

488.7 

2416.5 

44 

46 

4363.9 

431.7 

466.9 

23.59.8 

4456.3 

450.8 

489.5 

2418.5 

46 

48 

4367.0 

432.4 

467.7 

2361  7 

4459.3 

451.5 

490.3 

2420.4 

48 

50 

4370.1 

433.0 

46»  4 

2363.7 

4462  4 

452.1 

491.0 

2422.4 

50 

52 

4373.2 

433.6 

469.1 

2365  6 

4465.5 

452.7 

491  8 

2424.4 

52 

54 

4376  2 

434.3 

469.9 

2367.6 

4468.6 

453  4 

492.5 

2426.3 

54 

56 

43'^9.3 

434.9 

470.6 

2369.5 

4471.6 

454 . 1 

493.3 

2428 . 3 

56 

5« 

4382.4 

435  b 

471.4 

23-^1.5 

4474.7 

454.8 

494.1 

2430  2 

58 

60 

4385  5 

436  2 

472.1 

2373.4 

4477.8 

4.55.5 

494.8 

2432.2 

60 

280 

IX.     FUNCTIONS  OF  A 

ONE  DEGREE  CURVE. 

/ 

46°                         1 

4 

7° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C 

M. 

E. 

T. 

0 

4477.8 

4.55 . 5 

494.8 

2432.2 

4569.7 

475.2 

518.3 

2491.5 

0 

2 

4480  9 

456.1 

495.6 

24.34.2 

4.572.7 

475.9 

519  0 

2493.4 

2 

4 

4-183.9 

456.8 

496.5 

2136.1 

4575.8 

476.5 

519.8 

2495.4 

4 

6 

4487.0 

457.1 

407.2 

2438.1 

4578.8 

477.2 

520.6 

2497.4 

6 

8 

4490.0 

458.1 

497.9 

2440.1 

4581.9 

477.8 

521.4 

2499.4 

8 

10 

4493.1 

458.7 

498.7 

2442.1 

4584.9 

478.5 

522.2 

2501.4 

10 

1-2 

4496.2 

459.4 

499.5 

2444.0 

4588.0 

479.2 

.523.0 

2503.4 

12 

14 

4499.2 

460.0 

500.3 

2446.0 

4591.0 

479.8 

523.8 

2505.4 

14 

16 

4502.3 

460.7 

501.0 

^448.0 

4594.1 

480.5 

524.6 

2507.3 

16 

18 

4505.4 

461.3 

501.8 

2449.9 

4597.1 

481.1 

525.4 

2509.3 

18 

20 

4508.4 

462.0 

502.6 

2451.9 

4600.2 

481.7 

.526.2 

2511.3 

20 

oo 

4511.5 

462.7 

503.4 

2453.9 

4603.2 

482.3 

527.0 

2513.3 

22 

34 

4514.6 

463.3 

504.1 

2155  9 

460G.3 

483.0 

527.8 

2515.3 

24 

26 

4517.6 

464.0 

504.9 

2457.8 

4609.3 

483.7 

528.6 

2517.3 

26 

28 

4520.7 

404.6 

505.7 

2459.8 

4612.4 

484.3 

529.4 

2519.3 

28 

30 

4.523.7 

405.3 

506.5 

2461.8 

4615.4 

485.0 

530.2 

2521.2 

30 

32 

4526.8 

406.0 

507.3 

2463.8 

4618.5 

485.7 

.5.31.0 

2523.2 

32 

34 

4529.9 

466.6 

508.0 

2465.7 

4621.5 

486.3 

531.8 

2525.2 

34 

36 

4532.9 

467.3 

508.8 

2467.7 

4624.6 

487.0 

532.6 

2527.2 

36 

88 

4536.0 

467.9 

509.6 

2469.7 

4627.6 

487.7 

533.4 

2529.2 

38 

40 

45.39.1 

468.6 

510.4 

2471.7 

4630.7 

4S8.4 

5.34.2 

2531.2 

40 

42 

4542.1 

469.3 

511.1 

2473.6 

403:3.7 

489.1 

535.0 

2533.2 

42 

44 

4545.2 

469.9 

511.9 

2475.6 

4636.8 

489.8 

5.35.8 

2535.2 

44 

46 

4.548.2 

470.6 

512.7 

2477.6 

4639.8 

490.5 

536.6 

2537.2 

46 

48 

4551.3 

471.2 

513.5 

2479.6 

4642.9 

491.2 

537.4 

2539.2 

48 

50 

4554.4 

471.9 

514.3 

2481.6 

4645.9 

491.9 

538.2 

2541.2 

50 

52 

4557.4 

472.6 

515.1 

2483.5 

4649.0 

492.6 

5.39.0 

2543.1 

52 

54 

4560.5 

473.2 

515.9 

2485.5 

4652.0 

493.3 

539.8 

2.545.1 

54 

56 

4563.6 

473.9 

516.7 

2487.5 

46.55.1 

494.0 

540.6 

2547.1 

56 

58 

4.566.6 

474.5 

517.5 

2489.5 

46.58.1 

494.7 

541.4 

2549.1 

58 

60 

4569.7 

475.2 

518.3 

2491.5 

4661.2 

495.4 

542.3 

2551 . 1 

60 

/ 

48°                           1 

49°                          1 

/ 

L.  C. 
4661.2 

M. 

495.4 

E. 
542.3 

T. 
2.551.1 

L.  C. 

M. 

E. 

T. 

0 

47.52.3 

515.9 

567.0 

2611.3 

0 

2 

46C4  2 

496.0 

543.1 

2.5.53.1 

4755 . 3 

516.5 

567.8 

2613.3 

o 

4 

4667.3 

496.7 

543.9 

2555.1 

4758.4 

517.2 

568.7 

2615.3 

4 

6 

4670.3 

497.4 

544.7 

2,5.57.1 

4761.4 

517.9 

569.5 

2617.3 

6 

8 

4673.3 

498.1 

545.5 

2559.1 

4764.4 

518.6 

570.3 

2619.3 

8 

10 

4676.4 

498.8 

546.4 

2.5G1.1 

4767.4 

519.3 

.571.2 

2621.4 

10 

12 

4679.4 

499.4 

547.2 

2563.1 

4770.5 

520.0 

572.0 

2623.4 

12 

14 

4682.5 

500.1 

.548.0 

2.565.1 

4773.5 

.520.7 

572.8 

2625.4 

14 

16 

4685.5 

500.8 

548.8 

2.507.1 

4776.5 

.521.4 

573.7 

2627.4 

16 

18 

4688.5 

501.5 

549.6 

2569.1 

4779.6 

522.1 

574.5 

2629.4 

18 

20 

4691.6 

502.2 

.5.50.5 

2571.1 

4782.6 

522.8 

575.3 

2631 .4 

20 

22 

4694.6 

502.8 

551.3 

2573.1 

4785.6 

523.5 

576.2 

2633.5 

22 

24  , 

4697.6 

503.5 

552.1 

2575.1 

4788.7 

524.2 

577.0 

2635.5 

24 

26 

4700.7 

504.2 

5.52.9 

2577.1 

4791.7 

524.9 

577.9 

2637.5 

26 

28 

4703.7 

504.9 

5.53.7 

2.579.1 

4794.7 

.525.6 

578.7 

2639.5 

28 

30 

4706.7 

505.6 

.554.6 

2581.1 

4797.7 

526.3 

579.6 

2641.5 

30 

32 

4709.8 

506.2 

5.55.4 

2583.1 

4800.8 

527.0 

580.4 

2643.5 

32 

34 

4712.8 

506.9 

.556.2 

2585.1 

4803.8 

527.7 

581.3 

2645.6 

34 

S6 

4715.9 

507.6 

557.0 

2587.2 

4806.8 

528.4 

582.1 

2647.6 

36 

38 

4718.9 

508.3 

557.8 

2589.2 

4809.9 

529.1 

583.0 

2649.6 

38 

40 

4721.9 

509.0 

5.58.7 

2.591.2 

4812.9 

.529.8 

583.8 

2651.6 

40 

42 

4725.0 

509.6 

559.5 

2593.2 

4815.9 

530.5 

584.7 

26.53.7 

42 

44 

4728.0 

510.3 

560.3 

2595.2 

4819.0 

531.2 

585.5 

26.55.7 

44 

46 

4731.0 

511.0 

561.2 

2597.2 

4822.0 

531.9 

586.4 

2657.7 

46 

48 

4734.1 

511.7 

562.0 

2.599.2 

4825.0 

532.6 

.587.2 

2659.7 

48 

50 

4737.1 

512.4 

562.8 

2601.2 

4828.0 

.533.3 

588.1 

2661.8 

50 

52 

4740.2 

513.1 

563.7 

2603.2 

4831.1 

534.0 

588.9 

2663.8 

52 

54 

4743.2 

513.8 

564.5 

2605.2 

4834.1 

534.7 

589.8 

2665.8 

54 

56 

4746.2 

514.5 

565.3 

2607.2 

4837.1 

535.4 

590.6 

2667.8 

56 

58 

4749.3 

515.2 

566.2 

2609.3 

4S40.2 

536.1 

591.5 

2669.9 

58     ' 

60 

4752.3 

515.9 

567.0 

2611.3 

4843.2 

.536.8 

592.4 

2671.9 

60    1 

IX. -FUNCTIONS 

OF   A 

ONE-DEGREE  CURVE. 

281 

/ 

50°                      1 

51° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

4843.2 

536.8 

592.4 

2671. 9 

4933  6 

558.2 

618.5 

2733.0 

0 

o 

4846.2 

537.5 

593.2 

2673  9 

4936.6 

558.9 

619.3 

2735.1 

2 

4 

4849.2 

538.2 

594.1 

2676.0 

4939.6 

559.7 

620.2 

2737.1 

4 

6 

48.02.2 

538.9 

594.9 

2678.0 

4942.6 

560.4 

621.1 

2739.2 

6 

8 

4855.2 

539.6 

595.8 

2680.0 

4945.0 

561.1 

622.0 

2741.2 

8 

10 

4858.3 

540.3 

596  7 

2682.1 

4948.6 

561.8 

622.9 

2743.3 

10 

12 

4861.3 

541.0 

597.5 

2684.1 

4951.6 

562.5 

623.7 

2745.3 

12 

14 

4864.3 

541.7 

598.4 

2686.1 

49.54.6 

563.3 

624.6 

2747.4 

14 

16 

4867.3 

542.4 

599.3 

2688.2 

4957.6 

564.0 

625.5 

2749.4 

16 

18 

4870.3 

543.1 

600  1 

2690.2 

4960.6 

564.7 

626.4 

2751.5 

18 

20 

4873.3 

543.9 

601.0 

2692.3 

4963.6 

565.4 

627.3 

2753.5 

20 

22 

4876.3 

544.6 

601.9 

2694.3 

4966.6 

566.2 

628.2 

2755.6 

22 

24 

4879.4 

545.3 

602.7 

2696.3 

4969.6 

566.9 

629.9 

2757.7 

24 

26 

4882.4 

546.0 

603.6 

2698.4 

4972.6 

567.6 

630.0 

2759.7 

26 

28 

4885.4 

546.7 

604.5 

2700.4 

4975.6 

568.3 

630.9 

2761.8 

28 

30 

4888.4 

547.4 

605.3 

2702.4 

4978.6 

.569.1 

631.8 

2763.8 

30 

32 

4891.4 

548.1 

606.2 

2704.5 

4981.6 

569.8 

632.7 

2765.9 

32 

34 

4894.4 

548.8 

607.0 

2706.5 

4984.6 

570.5 

633.6 

2767.9 

34 

36 

4897.4 

549.5 

607.9 

2708.6 

4987.7 

571.2 

634.5 

2770.0 

36 

38 

4900.4 

550.2 

608.8 

2710.6 

4990.7 

572.0 

035.3 

2772.0 

38 

40 

4903.5 

551.0 

609.7 

2712.6 

4993.7 

572.7 

636.2 

2774.1 

40 

42 

4906.5 

551.7 

610.5 

2714.7 

4996.7 

573.4 

637.1 

2776.2 

42 

44 

4909.5 

552.4 

611.4 

2716.7 

4999.7 

574.1 

638.0 

2778.2 

44 

46 

4912.5 

553.1 

612.3 

2718.8 

.5002.7 

574.9 

638.9 

2780.3 

46 

48 

4915.5 

553.8 

613.2 

2720.8 

5005.7 

575.6 

639.8 

2782.3 

48 

50 

4918.5 

.554.5 

614.1 

2722.8 

5008.7 

576.3 

640.7 

2784.4 

50 

52 

4921.5 

555.2 

014.9 

2724.9 

.5011.7 

577.0 

641.6 

2786.4 

52 

54 

4924.6 

555.9 

615.8 

2726.9 

.5014.7 

577.8 

642.5 

2788.5 

54 

56 

4927.0 

556.6 

616.7 

2729.0 

5017.7 

578.5 

643.4 

2790.6 

56 

58 

4930.6 

557.4 

617.6 

2731.0 

5020.7 

579.2 

644.3 

2792.6 

58 

60 

4933.6 

558.2 

618.5 

2733.0 

5083.7 

579.9 

645.2 

2794.7 

60 

/ 

52°                          1 

5 

S° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

5023.7 

579.9 

645.2 

2794.7 

5113.5 

602.0 

672.7 

2856.9 

0 

»> 

.5026.7 

580.6 

6)6.1 

2796.8 

5116.5 

602.8 

673.7 

2858.9 

2 

4 

5029.7 

.581.3 

647.0 

2798.8 

5119.4 

603.5 

674.6 

2861.0 

4 

6 

.5032.7 

582.1 

647.9 

2800.9 

5122.4 

604.3 

675.5 

2863.1 

6 

8 

5035.7 

.582.8 

648.9 

2803.0 

5125.4 

605.0 

676.4 

2865.2 

8 

10 

5038.7 

583.5 

649.8 

2805.0 

5128.4 

605.8 

677.4 

2867.3 

10 

12 

5041.7 

584.3 

6.50.7 

2807.1 

5131.3 

606.5 

678.3 

2869.4 

12 

14 

5044.7 

585.0 

651.6 

2809.2 

5134.3 

607.3 

679.2 

2871.5 

14 

16 

5047.7 

585.7 

652.5 

2811.2 

5137.3 

608.0 

680.2 

2873.5 

16 

18 

5050.7 

.586.5 

653.4 

2813.3 

5140.3 

608.8 

681.1 

2875.6 

18 

20 

5053.6 

587.2 

654.3 

2815.4 

5143.2 

609.5 

682.0 

2877.7 

20 

22 

5056.6 

587.9 

6.55.2 

2817.4 

5146.2 

610.3 

683.0 

2879.8 

22 

24 

5059.6 

.588.7 

656.2 

2819.5 

5149.2 

611.0 

683.9 

2881.9 

.    24 

26 

5062.6 

.589.4 

6.57.1 

2821.6 

5152.1 

611.8 

684.9 

2884.0 

26 

28 

5065.6 

590.1 

6.58.0 

2823.6 

5155.1 

612.5 

685.8 

2886.1 

28 

30 

5068.6 

590.9 

6.58.9 

2825.7 

51.58.1 

613.3 

686.7 

2888.1 

30 

32 

5071.6 

591.6 

659.8 

2827.8 

5161.1 

614.0 

687.7 

2890.2 

32 

34 

5074.6 

592.3 

660.7 

2829.8 

5164.0 

614.8 

688.6 

2892.3 

34 

36 

5077.6 

593.1 

661.6 

2831.9 

5167.0 

615.5 

689.6 

2894.4 

36 

38 

5080.6 

593.8 

662.5 

2834.0 

5170.0 

616.3 

690.5 

2896.5 

38 

40 

5083.6 

594.5 

663.5 

2836.1 

5173.0 

617.0 

691.5 

2898.6 

40 

42 

5086.6 

595.3 

664.4 

2838.2 

5175.9 

617.8 

692.4 

2900.7 

42 

44 

5089.6 

596.0 

065.3 

2840.2 

5178.9 

61S.5 

693.4 

2902.8 

44 

46 

5092.6 

596.7 

666.2 

2842.3 

5181.9 

619.3 

694.3 

2904.9 

46 

48 

5095.6 

597.5 

667.2 

284.4.4 

5184.9 

620.1 

695.3 

2907.0 

48 

50 

5098.6 

598.2 

668.1 

2846.5 

5187.8 

620.8 

696.2 

2909.1 

50 

52 

5101.6 

598.9 

669.0 

2848.5 

5190.8 

621.5 

697.1 

2911.2 

52 

54 

5104.6 

.599.7 

669.9 

28.50.6 

5193.8 

622.3 

698.1 

2913.3 

54 

50 

5107.0 

600.4 

670.9 

2852.7 

5196.7 

623.0 

099.0 

2915.4 

56 

5S 

5110.6 

601.2 

671 .8 

2.S.^.4.S 

5199.7 

623.8 

700.0 

2917.5 

58 

60 

5113.5 

002.0 

672.7 

/- 

28.)6.9 

.520J.7 

624.6 

700.9 

2919.5 

60 

282 

IX.     FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 

1 

54 

OO"* 

/ 

L.  C. 

M. 

E. 

T. 

L.  C, 

M. 

E. 

T. 

0 

5202.7 

624.6 

700.9 

2919.5 

5291.7 

647.4 

7.9.9 

2982.8 

0 

2 

5205.7 

625  4 

701.9 

2921.6 

5294.6 

648.1 

730.9 

2984.9 

2 

4 

5208.6 

626.1 

70^.8 

2923.8 

5297.6 

648.9 

731.9 

2987.1 

4 

6 

5211.6 

6-,'6.9 

703.8 

2925.9 

5300.5 

649.6 

732.9 

2989.2 

6 

8 

5214.6 

627.6 

704.8 

2928.0 

5303.5 

650.4 

733.8 

2991.3 

8 

10 

5217.5 

628.4 

705.7 

2930.1 

5306.4 

651.2 

734.8 

2993.4 

10 

VI 

5220.5 

629.2 

706.7 

2932.2 

5309.4 

652.0 

735.8 

2995.5 

12 

14 

5223.5 

629.9 

707.7 

2934.3 

5312.3 

652.7 

736.8 

2997.7 

14 

16 

5226.4 

630.7 

708-6 

2936.4 

5315.3 

653.5 

737.8 

2999.8 

16 

18 

5229.4 

631.4 

709.6 

2938.5 

5318.2 

654.3 

738.7 

3001.9 

18 

20 

5232.4 

632.2 

710.5 

2940.6 

5321.2 

655.1 

739.7 

3004.0 

20 

22 

5235.3 

633.0 

711.5 

2942.7 

5.324.1 

655.8 

740.7 

3006.2 

22 

24 

5238.3 

633.7 

712.5 

2944.8 

5327.1 

656.6 

741.7 

3008.3 

24 

26 

5241.3 

634.5 

713.4 

2946.9 

5330.0 

657.4 

742.7 

3010.4 

26 

28 

5244.2 

635.2 

714.4 

2949.0 

5.3.33.0 

658.2 

743.7 

3012.5 

28 

30 

5247.2 

636.0 

715.3 

2951 . 1 

5.335.9 

658.9 

744.7 

3014.7 

30 

32 

5250.2 

636.8 

716.3 

2953.2 

5338.8 

659.7 

745  7 

3016.8 

a2 

34 

5253.1 

6.37.5 

717.3 

2955.3 

5341.8 

660.5 

746.7 

3018.9 

34 

36 

5256.1 

638.3 

718.2 

2957.5 

5344.7 

661.3 

747.7 

3021.1 

36 

38 

5259.1 

639.0 

719.2 

2959.6 

5347.7 

662.0 

748.7 

3023.2 

38 

40 

5262.0 

639.8 

720.2 

2961.7 

5350.6 

662.8 

749.7 

3025.3 

40 

42 

5265.0 

640.6 

721.1 

2963.8 

5353.6 

663  6 

7.50.7 

3027.5 

42 

44 

5268.0 

641.3 

722.1 

2965.9 

5356.5 

664.4 

751.7 

3029.6 

44 

46 

5270.9 

642.1 

723.1 

2968.0 

5359.5 

665.1 

752.6 

3031.7 

46 

48 

5273  9 

642.8 

724.1 

2970.1 

5362.4 

665.9 

753.6 

3033.8 

48 

50 

5276.9 

643.6 

725.0 

2972.2 

5365.4 

666.7 

754.6 

3036.0 

50 

52 

5279.8 

644.4 

726.0 

2974  4 

5368.3 

667.5 

755 . 6 

3038.1 

52 

54 

5282.8 

645.1 

727.0 

2976.5 

5371.3 

668.3 

756.6 

3040.2 

54 

56 

5285.8 

645.9 

728.0 

2978.6 

5374.2 

669.1 

7.57.6 

3042.4 

56 

58 

5288.7 

646.6 

729.0 

2980.7 

5377.2 

669.9 

758.6 

.3044.5 

58 

60 

5291.7 

647.4 

729  9 

2982.8 

5380.1 

670.7 

759.6 

3046.6 

60 

/ 

oG°                         1 

5 

;° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

5380.1 

670.7 

759.6 

30 16. 6 

5468.2 

694.4 

790.2 

3111.1 

0 

2 

5383.0 

671.4 

760.6 

3048.8 

5471.1 

695.2 

791.2 

3113.3 

2 

4 

5386.0 

672.2 

761.6 

3050.9 

5474.0 

696.0 

792.2 

3115.4 

4 

6 

5388.9 

672.9 

762.7 

30-3.1 

5477.0 

696.8 

793.3 

3117.6 

6 

8 

5391.8 

673.7 

763.7 

30.55.2 

5479.9 

697.6 

794.3 

3119.7 

8 

10 

5394.8 

674.4 

764.7 

3057.4 

548:i.8 

698.4 

795.3 

3121.9 

10 

12 

5397.7 

675.2 

765.7 

3059.5 

5485.7 

699.2 

796.3 

3124.1 

12 

14 

5400.7 

676.0 

766.7 

3061.6 

5488.7 

700.0 

797.4 

3126.2 

14 

16 

5403.6 

676.8 

767.7 

3063.8 

5491.6 

700.8 

798. 4 

3128.4 

16 

18 

5406.5 

677.6 

768.7 

3065.9 

.5494.5 

701.6 

799.4 

3130.6 

18 

20 

5409.5 

678.4 

769.7 

3068.1 

.5497.4 

702.4 

800.5 

3132.7 

20 

22 

5412.4 

679.2 

770.8 

3070.2 

.5500.3 

703.2 

801.5 

3134.9 

22 

24 

5415.3 

680.0 

771.8 

3072.4 

.5503.3 

704.0 

802.6 

3137.0 

24 

26 

5418.3 

680.8 

772.8 

3U74.5 

.5506.2 

704.8 

803.6 

3139.2 

26 

28 

5421.2 

681.6 

773.8 

3076.6 

5509.1 

705.6 

804.7 

3141.4 

28 

30 

5424.1 

682.4 

774.8 

3078.8 

5512.0 

706.4 

805.7 

3143.5 

30 

32 

5427.1 

683.2 

775.8 

3080.9 

5515.0 

707.2 

806.8 

3145.7 

32 

34 

5430.0 

684.0 

776.8 

3083.1 

5517.9 

708.0 

807.8 

3147.9 

34 

36 

54^3.0 

684.8 

777.8 

3085.2 

5520.8 

708.8 

808.8 

3150.0 

36 

38 

5435.9 

6S5.6 

778.9 

3087.4 

5523.7 

709.6 

809.9 

3152.2 

38 

40 

5438.8 

686.4 

779.9 

3089.8 

5.5-26.7 

710.4 

810.9 

3154.4 

40 

42 

5441.8 

687.2 

780. 9 

3091.7 

5.529.6 

711.2 

812.0 

3)56.6 

42 

44 

5444.7 

688.0 

781.9 

3093.9 

55.32.5 

712.0 

813.0 

31.58.7 

44 

46 

5447.6 

688.8 

783.0 

3096.0 

5535.4 

712.8 

814.1 

3160.9 

46 

48 

5450.6 

689.6 

784.0 

3098.2 

5538.4 

713.6 

815.1 

3163.1 

48 

50 

5453.5 

690.4 

785.0 

3100.3 

5541.3 

714.4 

816.2 

3165.3 

50 

52 

5456.5 

691.2 

786.0 

3102.5 

5544.2 

715.2 

817.2 

3167.4 

52 

54 

5459.4 

692.0 

787.1 

3104.6 

.5547.1 

716.0 

818.3 

3169.6 

54 

56 

54G2.3 

692.8 

788.1 

3106.8 

55.50.0 

716.8 

819.3 

3171.8 

56 

58 

54G.T.3 

093.6 

789.1 

3108.9 

.5553.0 

717.6 

820.4 

3174.0 

58 

60 

54()S.2 

m\.\ 

70:1.2 

3111.1 

5.5.55.9 

71S.4 

821.4 

3176.1 

60 

IX.— FUNCTIONS  OF  A  ONE  DEGREE  CURVE.     283 


/ 

o 

8° 

69° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

5555.9 

718.4 

821.4 

3176.1 

5643.1 

742.8 

8.53  5 

3241.9 

0 

o 

5558.8 

719  2 

822.5 

3178.3 

5646.0 

743.6 

8-A.G 

3244  I 

2 

4 

5561.7 

720.0 

823.5 

3180.5 

5648.9 

744.4 

855.7 

3246.3 

4 

6 

5564.6 

720.8 

824.6 

3182.7 

5651.8 

745.3 

856.8 

3248.5 

6 

8 

5567 . 5 

721.6 

825.7 

3184.9 

5054.7 

746.1 

8.57.9 

3250.7 

8 

10 

5570.4 

722.4 

826.7 

3187.1 

5657.6 

746.9 

859.0 

3252.9 

10 

12 

5573.3 

723.2 

827.8 

3189.2 

•5060.5 

747.7 

860.0 

3255.1 

12 

14 

.5576.2 

724.0 

828.9 

3191.4 

5663.4 

748.6 

861.1 

3257.3 

14 

16 

5579.2 

724.8 

829.9 

3193.6 

5666.3 

749.4 

862.2 

3259.5 

16 

18 

5582.1 

725.0 

831.0 

3195.8 

5669.2 

750.2 

863.3 

3261.7 

18 

20 

5585 ;0 

726.5 

832.1 

3198.0 

5672.1 

751.1 

864.4 

3263.9 

20 

22 

5587.9 

727.3 

833.1 

3200,2 

5675.0 

751.9 

865.5 

3266.1 

22 

24 

5590.8 

728.1 

834.2 

3202.4 

5677.9 

752.7 

866.6 

3268.3 

24 

26 

5593.7 

728.9 

835.3 

3204.5 

5680.8 

753.5 

867.7 

3270.5 

26 

28 

5596.6 

729.7 

836.3 

3206.7 

5683.7 

7.54.4 

868.8 

3272.7 

28 

30 

5599.5 

730  5 

837.4 

3208.9 

5086.5 

755  2 

869.9 

3274.9 

30 

32 

5602.4 

731.3 

838.4 

3211.1 

5689.4 

756.0 

871.0 

3277.1 

32 

34 

5605.3 

732.1 

839.5 

3213.3 

5692.3 

756  9 

872.1 

3279.4 

34 

36 

5608.2 

732.9 

840.6 

3215.5 

5695.2 

757.7 

873.2 

3281  6 

36 

38 

5611.1 

733.7 

841.0 

3217.7 

5698.1 

758.5 

874.3 

3283.8 

38 

40 

5614  0 

734.6 

842.7 

3219.9 

5701.0 

759.4 

875.4 

3286.0 

40 

42 

5616.9 

735.4 

843.8 

3222.1 

5703.9 

760.2 

876.5 

3288.2 

42 

44 

5619.8 

736.2 

844.9 

3224.3 

5706.8 

761.0 

877.6 

3290.5 

44 

46 

5622.8 

737.0 

846.0 

3226.5 

5709  7 

761.9 

878.7 

3292.7 

46 

48 

5625.7 

737.8 

847.0 

3228.7 

5712.6 

762.7 

879.8 

3294.9 

48 

50 

5628.6 

738.6 

848.1 

3230.9 

5715.5 

763.5 

880.9 

3297.1 

50 

52 

5631.5 

739.4 

849.2 

3233.1 

5718.4 

764.4 

882.0 

3299.3 

52 

54 

5634.4 

740.2 

850.3 

3235.3 

5721.3 

765.2 

883.1 

3301  5 

54 

56 

5637.3 

741  0 

851.4 

3237.5 

5724.2 

766.0 

884.2 

3303.8 

56 

58 

5640.2 

741.9 

852.5 

3239.7 

5727.1 

766.8 

885.3 

3306.0 

58 

60 

*■ 

5643.1 

742.8 

853.5 

3241.9 

5730.0 

767.7 

886.4 

3308.2 

60 

/ 

60°          1 

61° 

/ 

L.  C. 

M, 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

5730.0 

767.7 

886.4 

3308.2 

5816.4 

792.9 

920.2 

3375.2 

0 

2 

5732.9 

768.5 

887.5 

3310.4 

5819.3 

793.7 

921.4 

3377.4 

2 

4 

5735.8 

769.4 

888.7 

3312.7 

5822.1 

794.6 

922.5 

3379.7 

4 

6 

5738.6 

770.2 

889.8 

3314.9 

5825.0 

795.4 

923.6 

3381.9 

6 

8 

5741.5 

771.1 

890.9 

3317.1 

5827.9 

796.3 

924.8 

3384.2 

8 

10 

5744.4 

771.9 

892.0 

3319.3 

5830.7 

797.1 

925.9 

3386.4 

10 

12 

5747.3 

772.7 

893.1 

3321.6 

5833.6 

798.0 

927.1 

3388.7 

12 

14 

5750.2 

773.6 

894.3 

3323.8 

5836.5 

798.8 

928.2 

3390.9 

14 

16 

5753.0 

774.4 

895.4 

3326.0 

5839.3 

799.7 

929.3 

3393. S 

16 

18 

5755.9 

775.3 

896.5 

3328.3 

5842.2 

800.5 

930.5 

3395.4 

18 

20 

5758.8 

776.1 

897.6 

3330.5 

5845.1 

801.4 

931.6 

3397.7 

20 

22 

5761.7 

776.9 

898.8 

3332.7 

5847.9 

802.2 

932.8 

3399.9 

22 

24 

5764.6 

777.8 

899.9 

3334.9 

5850.8 

803.1 

933.9 

3402.2 

24 

26 

5767.4 

778.6 

901.0 

3337.2 

5853.7 

803.9 

935.1 

3404.4 

26 

28 

5770.3 

779.5 

902.1 

3339.4 

5856.5 

804.8 

936.3 

3406.7 

28 

30 

5773.2 

780.3 

903.2 

3341.6 

5859.4 

805.6 

937.4 

3408.9 

30 

32 

5776.1 

781.1 

904.4 

3343.9 

5862.3 

806.5 

938.6 

3411.2 

32 

34 

5779.0 

782.0 

905.5 

3346.1 

5865.1 

807.3 

939.7 

3413.5 

34 

36 

5781.8 

782.8 

906.6 

3348.3 

5868.0 

808.2 

940.9 

3415.7 

36 

38 

5784.7 

783.7 

907.7 

3350.6 

5870.9 

809.0 

942.1 

3418.0 

38 

40 

5787.6 

784.5 

908.8 

3352.8 

5873.7 

809.9 

943.2 

3420.3 

40 

42 

5790.5 

785.3 

910.0 

3355.0 

5876.6 

810.7 

944.4 

3422.5 

42 

44 

5793.4 

786.2 

911.1 

3357.3 

5879.5 

811.6 

945.5 

3424.8 

44 

46 

5796.2 

787.0 

912.3 

33.59.5 

5882. 3 

812.4 

946.7 

3427.1 

46 

48 

5799.1 

787.9 

913.4 

3361.8 

5885.2 

813.3 

947.8 

3429.3 

48 

50 

5802.0 

788.7 

914.5 

3364.0 

5S8S.1 

814.1 

949.0 

3431.6 

50 

52 

5804.9 

789.5 

915.7 

3366.2 

.5890,9 

815.0 

9.50.2 

3433.9 

52 

54 

5807.8 

790.4 

916.8 

3368.5 

.5893.8 

815.8 

951.3 

3436.1 

54 

56 

.5810.6 

791.2 

918.0 

.3370.7 

5S!)6 . 7 

816.7 

9.52.5 

3438.4 

56 

58 

5813.5 

792.1 

919.1 

3373.0 

.5899.5 

817.5 

9.53.6 

3440.7 

58 

60 

5816.4 

792.9 

920.2 

3375.2 

5902.4 

818.4 

954.8 

3442.9 

60 

N 

284     IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


1 

6 

2° 

63»           1 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

f 

0 

.-902.4 

818.4 

954.8 

3442.9 

5987. 8 

844.4 

990.3 

3511.3 

0 

2 

5905.2 

819.3 

9.56.0 

:3445.2 

5990.6 

845.3 

991.5 

3513.6 

2 

4 

5908.1 

820.1 

957.2 

3447.5 

5993.5 

846.2 

992.7 

3515.9 

4 

6 

5910.9 

821.0 

958  3 

:3449.7 

.5998.3 

847.1 

993.9 

3518.2 

6 

8 

5913.8 

821.8 

9.59.5 

:34.52.0 

5999.1 

847.9 

995.1 

3520.5 

8 

10 

5916.6 

822.7 

960.7 

.34.54.3 

6002.0 

848.8 

996.3 

3522.8 

10 

12 

5919.5 

823.6 

961.9 

.34.56 .« 

6004.8 

849.7 

997.5 

3.525.1 

12 

14 

5922.3 

824.4 

963.0 

3458.8 

C007.7 

850.6 

998.7 

3.527.4 

14 

16 

5925.2 

825.3 

9G4.2 

:3461.1 

0010.5 

851.4 

999.9 

3.529.7 

16 

18 

5928.0 

826.1 

965.4 

:3463.4 

6013.3 

852.3 

1001.1 

3532.0 

18 

20 

5930.9 

827.0 

966.6 

^465. 7 

0016.2 

853.2 

1002.3 

35:34.3 

20 

22 

.5933.7 

827.9 

967.8 

31G7.9 

0019.0 

854.1 

1003.5 

35.36.6 

22 

24 

5936.6 

828.7 

968.9 

3470.2 

6021.8 

854.9 

1004.7 

3.538.9 

24 

26 

.5939.4 

829.  G 

970.1 

3472.5 

6024.7 

855.8 

1005.9 

3541.2 

26 

28 

59-12.3 

830.4 

971.3 

3474.7 

G027.5 

8.56.7 

1007.1 

.3543.5  ' 

28 

30 

5945.1 

831.3 

972.5 

3477.0 

0030. 3 

857.6 

1008.4 

.3545.8  , 

30 

32 

5947.9 

832.2 

973.6 

3479.3 

GO.33.2 

858.4 

1009.6 

3548.1  : 

32 

34 

.5950.8 

833.0 

974.8 

3481.6 

00.36.0 

859.3 

1010.8 

3550.4  1 

34 

36 

5953.6 

833.9 

976.0 

;3483.9 

60:38.9 

860.2 

1012.0 

a552.7 

36 

38 

5956.5 

834.7 

977.2 

3486.2 

0041.7 

861.1 

1013.2 

3555.0 

38 

40 

.5959.3 

8:35.6 

978.4 

3488.5 

G044.5 

861.9 

1014.5 

3557.3 

40 

42 

5962.2 

83G.5 

979.6 

3490.7 

0047.4 

862-8 

1015.7 

35.59.6 

42 

44 

5965.0 

837.4 

980.8 

;3493.0 

6050.2 

863.7 

1016.9 

3562.0 

44 

46 

5967.9 

838.3 

982.0 

:3495.3 

0053.0 

864.6 

1018.1 

;3564.3 

46 

48 

5970.7 

8:39.1 

983.2 

3497.6 

00.55.9 

865.4 

1019.3 

3.566.6 

48 

50 

5973.6 

840.0 

984.4 

3499.9 

60.5^^.7 

866.3 

1020.6 

3568.9 

50 

52 

5976.4 

840.9 

985.5 

:3502.2 

00G1.6 

867.2 

1021.8 

3571.2 

52 

54 

.5979.3 

841.7 

9S6.7 

3.504.5 

6004.4 

868.1 

1023.0 

.3573.5 

54 

56 

5982.1 

842.6 

9s7.9 

;3506.S 

0067.2 

868.9 

1024.2 

:3575.8 

56 

58 

5985.0 

843.5 

989.1 

3.509.0 

0070.1 

869.8 

1025.4 

3578.1 

58 

60 

5987.8 

844.4 

990.3 

3511.3 

0072.9 

870.7 

1026.7 

3580.4 

60 

/ 

64°,          1 

t 

.5° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

6072.9 

870.7 

1026.7 

3.580.4 

6157.5 

897.3 

1064.0 

36.50.4 

0 

2 

6075.7 

871.5 

1027.9 

;35S2.S 

0160.3 

898.2 

1065.2 

36.52.8 

2 

4 

6078.5 

872.4 

1029.2 

3585.1 

6103.1 

899.1 

1066.5 

.3655.1 

4 

6 

6081.4 

873.3 

10:30.4 

3587.4 

6165.9 

900.0 

1067.7 

36.57.5 

6 

8 

6084.2 

874.2 

1031.7 

3589.7 

0168.7 

900.9 

1069.0 

3659.8 

8 

10 

6087.0 

875.1 

10:32.9 

3.592.1 

6171.5 

901.8 

1070.2 

3662.2 

10 

12 

6039.8 

875.9 

1034.1 

:3594.4 

6174.3 

902.7 

1071.5 

3664.5 

12 

14 

6092.6 

876.8 

10:35.4 

3596.7 

6177.1 

903.6 

1072.7 

3666.9 

14 

16 

6095.5 

Oi  1  .  t 

10:36.6 

:3599.1 

6179.9 

904.5 

1074.0 

3669.2 

16 

18 

6098.3 

878.6 

1037.9 

3601.4 

6182.7 

905.4 

1075.2 

3671. G 

18 

20 

6101.1 

879.5 

1039.1 

3603.7 

6185.5 

906.3 

1076.6 

3673.9 

20 

22 

6103.9 

880.3 

1040.3 

3606.0 

6188.3 

907.2 

1077.8 

3676.2 

22 

24 

6106.7 

881.2 

1041.6 

3608. 4 

6191.1 

908.1 

1079.1 

:3678.6 

24 

26 

6109.6 

882.1 

1042.8 

3610.7 

6193.9 

909.0 

1080.4 

3680.9 

26 

28 

6112.4 

883.0 

1044.1 

3613.0 

6196.7 

909.9 

1081.7 

3683.3 

28 

30 

6115.2 

88:3.9 

1045.3 

3615.3 

6199.5 

910.8 

1083.0 

3685.6 

30 

32 

6118.0 

884.7 

1046.5 

:3617.7 

6202.3 

911.7 

1084.2 

3688.0 

32 

34 

6120.8 

885.6 

1047.8 

3620.0 

6205.1 

912.6 

1085.5 

3690.4 

U 

86 

6123.7 

886.5 

1049.0 

3622.3 

6208.0 

913.5 

1086.8 

3692.7 

36 

38 

6126.5 

887.4 

10.50.3 

3624.7 

6210.8 

914.4 

1088.1 

:3695.1 

38 

40 

6129.3 

888.3 

1051.5 

:3627.0 

6213.6 

915.3 

1089.4 

.3697.4 

40 

42 

6132.1 

889.2 

10.52.7 

.3G29.4 

6216.4 

916.2 

1090.0 

36&9.8 

42 

44 

6i:i4.9 

890.1 

10,54.0 

:3631.7 

6219.2 

917.1 

1091.9 

.3702.2 

44 

46 

6137.8 

891.0 

1055.2 

:36:34.0 

6222.0 

918.0 

1093.2 

.3704.5 

46 

48 

6140.6 

891.9 

10.56.5 

36:3G.4 

6224.8 

918.9 

1094.5 

3706.9 

48 

50 

6143.4 

892.8 

10.57.7 

.3638.7 

6-227.6 

919.8 

1095.8 

3709.3 

50 

52 

6146.2 

893.7 

10.59.0 

3641.1 

6230.4 

920.7 

1097.0 

•3711.6 

52 

54 

6149.0 

894.6 

1060.2 

364:^.4 

62:33.2 

921.6 

109P.3 

3714.0 

54 

56 

6151.9 

895.5 

1061.5 

3645.7 

62.36.0 

922.5 

1099.6 

3716.3 

56 

58 

6154.7 

896.4 

10G2.7 

3648.1 

02.38.8 

923.4 

1100.9 

:37]S.7 

58 

60 

6157.5 

897.. S 

10G4.0 

36.50.4 

6241.6 

924.3 

1102.2 

.3721.1 

1  60 

IX.— FUNCTIONS   OF  A  ONE-DEGREE   CURVE.    285 


/ 

66 

o 

67 

o 

/ 

L.  C.   M. 

E. 

T. 

L.  C, 

M. 

E. 

T. 

0 

6241.6  924.3 

1102.2 

3721.1 

6325.2 

951.8 

1141.5 

3792.6 

0 

2 

6244.4  925.2 

1103.5 

3723.4 

6328.0 

952.7 

1142.8 

3795.0 

2 

4 

6247.2  9-26.1 

1104.8 

37-^5.8 

6330.7 

953.6 

1144.1 

3797.4 

4 

6 

6250.0  927.0 

1106.1 

3728.2 

6333.5 

954.5 

1145.4 

3799.8 

6 

8 

6252.7  927.9 

1107.4 

3730.6 

6330.3 

955.5 

1146.7 

3802.2 

8 

10 

6255.5  928.8 

1108.7 

3732.9 

6339.0 

950.4 

1148.1 

3804.6 

10 

12 

6258.3  929.8 

1110. 0 

3735.3 

0341.8 

957.3 

1149.4 

3807.0 

12 

14 

6261.1  930.7 

1111.3 

3737.7 

6344.6 

958.2 

1150.7 

3809.4 

14 

16 

6263.9  931  6 

1112.6 

3740.1 

6347.4 

059.2 

1152.0 

3811.8 

16 

18 

6260. 7  932.5 

1113.9 

3742.4 

6350.1 

900.1 

1153.3 

3814.2 

18 

20 

6269.5  '933.4 

1115.2 

3744.8 

6352.9 

901.0 

1154.7 

3816.6 

20 

22 

6272.3  934.3 

1116.5 

3747.2 

6355.7 

961.9 

1156.0 

3819.0 

22 

24 

6275  0  935.3 

1117.8 

3749.0 

0358.4 

962.9 

1157.4 

3821.4 

24 

26 

6277.8  936.2 

1119.1 

3751.9 

6361.2 

963.8 

1158.7 

3823.8 

26 

28 

62S0.6  937.1 

1120.4 

3754.3 

6364.0 

904.7 

1160.1 

3826.2 

28 

30 

6283.4  938.0 

1121.7 

3756.7 

6306.7 

905.6 

1161.4 

3828.6 

30 

32 

6286.2  938.9 

1123.0 

3759.1 

6309.5 

960.6 

1162.8 

3831 .0 

32 

34 

6289.0  939.8 

1124.3 

3701.5 

0372.3 

907.5 

1164.1 

3833.4 

34 

36 

6291.8  940.8 

1125.6 

3703.9 

6375.1 

968.4 

1165.5 

3835.9 

36 

38 

6294.5  941.7 

1126.9 

3766.3 

6377.8 

909.3 

1166.8 

3838.3 

38 

40 

6297.3  942  6 

1128.3 

3708.7 

6380.6 

970.3 

1168.2 

3840.7 

40 

42 

6300.1  943.5 

1129.6 

3771.0 

6383.4 

971.2 

1169.5 

3843.1 

42 

44 

6302.9  944.4 

1130.9 

3773.4 

6380.1 

972.1 

1170.9 

3845.5 

44 

46 

6305.7  915.3 

1132.2 

3775.8 

6388.9 

973.0 

1172.2 

3847.9 

46 

48 

6308.5  946.3 

1133.5 

3778.2 

6391.7 

974.0 

1173.6 

3850.4 

48 

50 

6311.3  947.2 

1134.9 

3780.6 

6394.4 

974.9 

1174.9 

3852.8 

50 

52 

6314.1  948.1 

1136.2 

3783.0 

6397.2 

975.8 

1176.3 

3855.2 

52 

54 

6316.8  949.0 

1137.5 

3785.4 

6400.0 

976.8 

1177.6 

3857.6 

54 

56 

6319.6  949.9 

1138.8 

3787.8 

6402.8 

977.7 

1179.0 

3860.0 

56 

58 

6322.4  950.8 

1140.1 

3790.2 

6405.5 

978.6 

1180.3 

3862.5 

58 

60 

6325.2  951.8 

1141.5 

3792.6 

6408.3 

979.6 

1181.6 

3864.9 

60 

/ 

1 

6S 

»° 

6S 

•° 

/ 

L.  C.   M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

6108.3  979.6 

1181.6 

3864.9 

6491.1 

1007.7 

1222.9 

3938.1 

0 

2 

6411.1  980.5 

1183.0 

3867.3 

6493.8 

1008.7 

1224.3 

3940.6 

2 

4 

6413.8  981.4 

1184.4 

3869.7 

6496.6 

1009.6 

1225.7 

3943.0 

4 

6 

G416.6  982.4 

1185.7 

3872.2 

6499.3 

1010.6 

1227.1 

3945.5 

6 

8 

6419.3  983.3 

1187.1 

3874.6 

6502.1 

1011.5 

1228.5 

3947.9 

8 

10 

6422.1  984.2 

1188.5 

3877.0 

6504.8 

1012.5 

1229.9 

3950.4 

10 

12 

642 1.9  985.2 

1189.8 

3879.5 

6.507.5 

1013.4 

1231.3 

3952.9 

12 

14 

6427. 6  986.1 

1191.2 

3881.9 

6510.3 

1014.4 

1232.7 

3955.3 

14 

16 

6430.4  987.0 

1192.6 

3884.3 

6513.0 

1015.3 

1234.1 

3957.8 

16 

18 

6433.1  988.0 

1193.9 

3886.8 

6515.8 

1016.3 

1235.5 

3960.2 

18 

20 

64a5.9  988.9 

1195.3 

3889.2 

6518.5 

1017.2 

1236.9 

3962.7 

20 

22 

6438.7  989.8 

1196.7 

3891.6 

6521.2 

1018.2 

1238.3 

3965.2 

22 

24 

6441.4  990.8 

1198.0 

3894.1 

6524.0 

1019.1 

1239.7 

3967.6 

24 

26 

6444.2  991.7 

1199.4 

3896.5 

6520.7 

1020.1 

1241.1 

3970.1 

26 

28 

6446.9  992.6 

1200.8 

3898.9 

0529.5 

1021.0 

1242.5 

3972.5 

28 

30 

6449.7  993.6 

1202.1 

3901.4 

6532.2 

1022.0 

1243.9 

3975.0 

30 

32 

6452.5  994.5 

1203.5 

3903.8 

6534.9 

10J2.9 

1245.3 

3977.5 

32 

34 

6455.2  995.4 

1204.9 

3900.3 

6537.7 

1023.9 

1246.7 

3980.0 

34 

36 

6458.0  996.4 

1200. 2 

3908.7 

6.540.4 

1024.8 

1248.1 

3982.4 

36 

38 

6460.7  997.3 

1207.6 

3911.2 

6543.2 

1025.8 

1249.5 

3984.9 

38 

40 

6403.5  998.2 

1209.0 

3913.6 

6545.9 

1020.7 

12.50.9 

3987.4 

40 

42 

6466.3  999.2 

1210.3 

3916.1 

6548.6 

1027.7 

1252.3 

3989.9 

42 

44 

6469.0  1000.1 

1211.7 

3918.5 

0551.4 

1028.6 

12.53.7 

3992.3 

44 

46 

6471.8  1001.0 

1213.1 

3921.0 

6554.1 

1029.6 

1255.1 

3994.8 

46 

48 

6474.5  1002.0 

1214.5 

3923.4 

6556.9 

1030.5 

1256.5 

3997.3 

48 

50 

6477.3  1002.9 

1215.9 

3925.9 

6559.6 

1031.5 

12.57.9 

3999.8 

50 

52 

6480.1  1003.8 

1217.3 

3928.3 

6562.3 

1032.4 

1259.3 

4002.2 

52 

54 

6482.8  1004.8 

1218.7 

39.30.8 

0.505.1 

1033.4 

1260.7 

4004.7 

54 

56 

6485.6  1005.7 

1220.1 

39.33.2 

6507.8 

1034.3 

1262.1 

4007.2 

56 

58 

6488.3  1006.7 

1221.5 

3935.7 

0.570.6 

1035  3 

1263.5 

4009.7 

58 

1  60 

6491.1  1007.7 

1222.9 

3938.1 

6573.3 

1036.3 

1265.0 

4012.1 

60 

286    IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


70° 

71 

[° 

0 

L.  C. 

6573.3 

M. 

1036.3 

E. 

1265.0 

T. 
4012.1 

L.  C. 

M. 

E. 

T. 

/ 

6654.9 

1065.1 

1308.4 

4087.1 

0 

2 

6576.0 

1037.3 

1266.4 

4014.6 

6657.6 

1066.1 

1309.9 

4089.7 

2 

4 

6578.7 

1038.2 

1267.9 

4017.1 

6660.3 

1067.0 

1311.3 

4092.2 

4 

6 

6581.5 

1039.2 

1269.3 

4019.6 

6663.0 

1068.0 

1312.8 

4094.7 

6 

8 

6584.2 

1040.1 

1270.8 

4022.1 

6665.7 

1068.9 

1314.2 

4097.2 

8 

10 

6586.9 

1041.1 

1272.2 

4024.6 

6668.4 

1069.9 

1315.7 

4099.8 

10 

\2 

6589.6 

1042.1 

1273.6 

4027.1 

6671.1 

1070.9 

1317.2 

4102.3 

12 

14 

6.592.3 

1043.0 

1275.1 

4029.6 

6673.8 

1071.9 

1318.6 

4104.8 

14 

16 

6595.1 

1044.0 

1276.5 

4032.1 

6676.6 

1072.9 

1320.1 

4107.3 

16 

18 

6597.8 

1044.9 

1278.0 

4034.6 

6679.3 

1073.8 

1321.5 

4109.8 

18 

20 

6600.5 

1045.9 

1279.4 

4037.1 

6682.0 

1074.8 

1323.0 

4112.4 

20 

22 

6603.2 

1046.9 

1280.8 

4039.6 

6684.7 

1075.8 

1324.4 

4114.9 

22 

24 

6605.9 

1047.8 

1282.3 

4042.1 

6687.4 

1076.8 

1325.9 

4117.4 

24 

26 

6608.7 

1048.8 

1283.7 

4044.6 

6690.1 

1077.7 

1327.4 

4119.9 

26 

28 

6611.4 

1049.7 

1285.2 

4047.1 

6692.8 

1078.7 

1.328.9 

4122.4 

28 

30 

6614.1 

1050.7 

1286.6 

4049.6 

6695.5 

1079.7 

1330.4 

4125.0 

30 

32 

6616.8 

1051.7 

1288.0 

40.52.1 

6698.2 

1080.7 

1331.8 

4127.5 

32 

34 

6619.5 

1052.6 

1289.5 

4054.6 

6700.9 

1081.6 

1333.3 

4130.4 

34 

36 

6622.3 

1053.6 

1290.9 

4057.1 

6703.6 

1082.6 

1.334.8 

4132.6 

36 

38 

6625.0 

1054.5 

1292.4 

4059.6 

6706.3 

1083.6 

1336.3 

4135.1 

38 

40 

6627.7 

1055.5 

1293.8 

4062.1 

6709.0 

1084.5 

1337.8 

4137.7 

40 

42 

6630.4 

1056.5 

1295.3 

4064.6 

6711.7 

1085.5 

1339.2 

4140.2 

42 

44 

6633.1 

1057.4 

1296.7 

4067.1 

6714.4 

1086.5 

1340.7 

4142.7 

44 

46 

6635.9 

1058  4 

1298.2 

4069.6 

6717.2 

1087.5 

1342.2 

4145.3 

46 

48 

6638.6 

1059.3 

1299.6 

4072.1 

6719.9 

1088.4 

1343.7 

4147.8 

48 

50 

6641.3 

1060.3 

1301.1 

4074.6 

6722.6 

1089.4 

1345.2 

4150.4 

50 

52 

6644. C 

1061.3 

1302.6 

4077.1 

6725.3 

1090.4 

1.346.7 

4152.9 

52 

54 

6646.7 

1062.2 

1304.0 

4079.6 

6728.0 

1091.3 

1348.2 

41.55.4 

54 

56 

6649.5 

1063.2 

1305.5 

4082.1 

6730.7 

1092.3 

1349.7 

4158.0 

56 

58 

66.52.2 

1064.1 

1306.9 

4084.6 

6733.4 

1093.3 

1351.2 

4160.5 

58 

60 

L . 

6654.9 

1065.1 

1308.4 

4087.1 

6736.1 

1094.3 

1352.7 

4163.1 

60 

/ 

72 

o 

73 

," 

0 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

6736.1 

1094  3 

13.52.7 

4163.1 

6816.6 

1123.9 

1398.1 

4240.0 

2 

6738.8 

1095  2 

1354.2 

4165.6 

6819.3 

1124.8 

1399.6 

4242.6 

2 

4 

6741.5 

1096.2 

1355.7 

4168.2 

6821.9 

1125  8 

1401.2 

4245.1 

4 

6 

6744.1 

1097.2 

1357.2 

4170.7 

6824.6 

1126.8 

1402.7 

4247.7 

6 

8 

6746.8 

1098.2 

1358.7 

4173.3 

6827.3 

1127.8 

1404.2 

4250.3 

8 

10 

6749.5 

1099.2 

1360.2 

4175.8 

6830.0 

1128.8 

1405.8 

4252.9 

10 

12 

6752.2 

1100.1 

1361.7 

4178.4 

6832.6 

1129.8 

1407.3 

4255.5 

12 

14 

6754.9 

1101.1 

1363.2 

4181.0 

68.35.3 

1130.8 

1408.8 

4258.1 

14 

16 

6757  6 

1102.1 

1.364.7 

4183.5 

6838.0 

1131.8 

1410.4 

4260.7 

16 

18 

6760.2 

1103.1 

1366.2 

4186.1 

6840.7 

1132.8 

1411.9 

4263.2 

18 

20 

6762.9 

1104.1 

1367.7 

4188.6 

6843.3 

1133.8 

1413.5 

4265.8 

20 

22 

6765.6 

1105.1 

1369.2 

4191.2 

6846.0 

1134  8 

1415.1 

4268.4 

22 

24 

6768.3 

1106.0 

1370.7 

4193.7 

6848.7 

11.35.8 

1416.6 

4271.0 

24 

26 

6771.0 

1107.0 

1372.2 

4196.3 

6851.3 

1136.8 

1418.2 

4273.6 

26 

28 

6773.7 

1108.0 

1373.7 

4198.8 

6854. 0 

1137.8 

1419.7 

4276.2 

28 

30 

6776.3 

11U9.0 

1.375.2 

4201.4 

68.56.7 

1138.8 

1421.3 

4278.8 

30 

32 

6779.0 

1109.9 

1376.7 

4204.0 

6859.4 

1139.8 

1422.9 

4281.4 

32 

34 

6781.7 

1110.9 

1378.2 

4206.5 

6862.0 

1140.8 

1424.4 

4284.0 

34 

36 

6784.4 

1111.9 

1379.7 

4209.1 

6864.7 

1141.8 

1426.0 

4286.6 

36 

38 

6787.1 

1112.9 

1381.2 

4211.7 

6867.4 

1142.8 

1427.5 

4289.2 

38 

40 

6789.8 

1113.9 

1382.8 

4214.3 

6870.1 

1143.8 

1429.1 

4291.8 

40 

42 

6792.4 

1114.9 

1384.3 

4216.8 

6872.7 

1144.8 

1430.7 

4294.4 

42 

44 

6795.1 

1115.9 

13S5.8 

4219.4 

6875.4 

1145.8 

1432.2 

4297.0 

44 

46 

6797.8 

1116.9 

1387.4 

4222.0 

6878.1 

1146.8 

1433.8 

4299.6 

46 

48 

6800.5 

1117.9 

1388.9 

4224.5 

6880.8 

1147.8 

1435.3 

4302.2 

48 

50 

6803.2 

1118.9 

1390.4 

4227.1 

6883.4 

1148  8 

1436.9 

4304.8 

50 

52 

6805.9 

1119.9 

1.392.0 

4229.7 

6886.1 

1149.8 

1438.5 

4307.4 

52 

54 

6808.5 

1120.9 

1393.5 

42.32.3 

6888.8 

1150.8 

1440.0 

4310.0 

54 

56 

6811.2 

1121.9 

1395.0 

4234.8 

6891.4 

1151.8 

1441.6 

4312.6 

56 

58 

6813.9 

1122.9 

1396.6 

42.37.4 

6894.1 

1152.8 

1443.1 

4315.2 

58 

60 

6816.6 

1123.9 

1398.1 

4240.0 

6896.8 

1153.8 

1444.7 

4317.8 

60 

IX.— fu:nctions  op  a  one-degree  curve.    287 


0 
2 

4 

6 

8 

10 

1^> 

14 
16 

18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 

42 
44 
46 
48 
50 
52 
54 
56 
58 
60 


'4° 


L.  C. 


M. 


E. 


T. 


6896.8 
6899.4 
6902.1 
6904.8 
6907.4 
6910.1 
6912.7 
6915.4 
6918.0 
6920.7 

6923.3 
6926.0 
6928.6 
6931.3 
6933.9 
6936.6 
6939.2 
6941.9 
6944.6 
6947.2 

6949.9 

6952.5 

6955.2 

6957.8 

6960.5 

6963.1 

6965.8 

6968.4 

6971.1. 

6973.7 

6976.4 


1153.8 
1154.8 
1155.8 
1156.8 
1157.8 
1158.8 
1159.8 
1160.8 
1161.8 
1162.8 

1163.9 
1164.-9 
1165.9 
1166.9 
1167.9 
1168.9 
1169.9 
1170.9 
1171.9 
1172.9 

1174.0 
1175.0 
1176.0 
1177.0 
1178.0 
1179.0 
1180.0 
1181.0 
1182.0 
1183.0 
1184.1 


1444.7 
1446.2 
1447.8 
1449.4 
1451.0 
1452.6 
1454.1 
1455.7 
1457.3 
1458.9 

1460.5 
1462.0 
1463.6 
1465.2 
1466.8 
1468.4 
1469.9 
1471.5 
1473.1 
1474.7 

1476.4 
1478.0 
1479.6 
1481.2 
1482.8 
1484.4 
1486.0 
1487.7 
1489.3 
1490.9 
1492.5 


4317.8 
43-'0.5 
4323.1 
4325.7 
4328.3 
4330.9 
4333.6 
4336.2 
4338.8 
4341.4 

4344.0 
4346.7 
4349.3 
4351.9 
4354.5 
4357.1 
4359.8 
4362.4 
4365.1 
4367.7 

4370.3 
4373.0 
4375.6 
4378.3 
4380.9 
4383.5 
4386.2 
4388.8 
4391.5 
4394.1 
4396.7 


to" 


L.  C. 


M. 


E. 


T. 


6976.4 
6979.0 
6981.7 
6984.3 


6986.9 

6989, 

6992. 

6994, 

6997, 


7000.1 

7002.8 
7005.4 
7008.0 
7010.7 
7013.3 
7015.9 
7018.6 
7021.2 
7023.9 
7026.5 

7029.1 
7031.8 
7034.4 
7037.0 
7039.7 
7042.3 
7045.0 
7047.6 
7050.2 
7052.9 
7055.5 


1184.1 
1185.1 
1186.1 
1187.1 
1188.1 
1189.2 
1190.2 
1191.2 
1192.2 
1193.2 

1194.3 
1195.3 
1196.3 
1197.3 
1198.3 
1199.4 
1200.4 
1201.4 
1202.4 
1203.4 

1204.5 
1205.5 
1206.5 
1207.5 
1208.5 
1209.6 
1210.6 
1211.6 
1212.6 
1213.6 
1214.7 


1492.5 
1494.1 
1495.7 
1497.3 
1499.0 
1500.6 
1.502.2 
1503.8 
1505.4 
1507.0 

1508.7 
1510.3 
1512.0 
1513.6 
1515.3 
1516.9 
1518.5 
1520.2 
1521.8 
1523.5 

1525.1 
1.526.7 
1528.4 
1530.0 
1531.7 
1533.3 
1534.9 
1.536.6 
1.538.2 
1539.9 
1541.5 


4396.7 
4399.4 
4402.1 
4404.7 
4407.4 
4410.0 
4412.7 
4415.3 
4418.0 
4420.7 

4423.3 
4426.0 
4428.6 
4431.3 
4434.0 
4436.6 
4439.3 
4442.0 
4444.6 
4447.3 

4450.0 
4452.7 
4455.3 
4458.0 
4460.7 
4463.4 
4466.0 
4468.7 
4471.4 
4474.1 
4476.7 


0 
2 
4 

6 
8 
10 
12 
14 
16 
18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 


7 

6" 

7 

70 

0 

L.  C. 

M. 

E. 

T. 

L.  C. 

M 

E. 

T. 

7055.5 

1214.7 

1541.5 

4476.7 

7134.0 

1245.6 

1.591.7 

4557.8 

0 

2 

7058.1 

1215.7 

1543.2 

4479.4 

7136.6 

1246.6 

1593.4 

4560.5 

2 

4 

7060.7 

1216.7 

1544.9 

4482.1 

7139.2 

1247.7 

1595.1 

4563.3 

4 

6 

7063.3 

1217.8 

1546  5 

4484.8 

7141.8 

1248.7 

1596.8 

4566.0 

6 

8 

7066.0 

1218.8 

1548.2 

4487.5 

7144.4 

1249.8 

1598.5 

4568.7 

8 

10 

7068.6 

1219.8 

1549.9 

4490.2 

7147.0 

12.50.8 

1600.2 

4571.5 

10 

12 

7071.2 

1220.9 

1551.5 

4492.9 

7149.6 

1251.8 

1601.9 

4574.2 

12 

14 

7073.8 

1221.9 

1553.2 

4495.6 

7152.2 

1252.9 

1603.6 

4576.9 

14 

16 

7076.4 

1222.9 

1554.9 

4498.3 

7154.8 

1253  9 

1605.3 

4579.7 

16 

18 

7079.0 

1224.0 

1556.5 

4.501.0 

7157.4 

1255  0 

1607.0 

4582.4 

18 

20 

7081.7 

1225.0 

1.558.2 

4.503.7 

7160.0 

12.56.0 

1608.7 

4585.1 

20 

22 

7084.3 

1226.0 

1.5.59.9 

4.506.  S 

7162.6 

12.57.0 

1610.4 

4587.9 

22 

24 

7086.9 

1227.1 

1561.5 

4509.0 

7165.2 

12.58.1 

1612.1 

4^0.6 

24 

26 

7089.5 

1228.1 

1.563.2 

4511.7 

7167.8 

1259.1 

1613.8 

4593.3 

26 

28 

7092.1 

1229.1 

1564  9 

4514.4 

7170.4 

1260.2 

1615.5 

4596.0 

28 

30 

7094.7 

1230.2 

1566.5 

4517.1 

7173.0 

1261.2 

1617.3 

4598  8 

30 

32 

7097.4 

1231.2 

1568.2 

4519.8 

7175.6 

1262.2 

1619.0 

4601.5 

32 

34 

7100.0 

1232.2 

1.569.9 

4522.5 

7178  2 

1263.3 

1620.7 

4604.3 

34 

36 

7102.6 

1233.3 

1571.5 

4525.3 

7180.8 

1264.3 

1622.4 

4607.0 

36 

38 

7105.2 

1234.8 

1573.2 

4528.0 

7183.4 

1265.4 

1624.1 

4609^ 

38 

40 

7107.8 

1235.3 

1.574.8 

4530.7 

7186.0 

1266.4 

1625.9 

4612.5 

40 

42 

7110.4 

1236.4 

1576.4 

4.533.4 

7188.6 

1267.4 

1627.6 

4615.3 

42 

44 

7113.1 

1237.4 

1578.1 

45,36,1 

7191.2 

1268.5 

1629.3 

4618.0 

44 

46 

7115.7 

1238  4 

1579.8 

4.538.8 

7193.8 

1269.5 

1631.0 

4620.8 

46 

48 

7118.3 

1239  5 

1.581 .5 

4.541.5 

7196.4 

1270.6 

1632.7 

4623.5 

48 

50 

7120.9 

1240  5 

1.583.2 

4544.2 

7199.0 

1271.6 

1634  5 

4626.3 

50 

52 

7123.5 

1241.5 

1584.9 

4.547.0 

7201.6 

1272.7 

1636.2 

4629.0 

52 

54 

7126  1 

1242.6 

1586.6 

4.549.7 

7204  2 

1273.7 

1637.9 

4631.8 

54 

56 

7128.8 

1243.6 

1588.3 

4,5.52.4 

7206.8 

1274.8 

1639.6 

4634.5 

56 

58 

7131.4 

1244.6 

1590  0 

4.5.55.1 

7209.4 

1275.8 

1641.3 

4637.3 

58 

60 

7134  0 

1215.6 

1591.7 

4.557.8 

7212.0 

1276.9 

1643.1 

4640.0 

60  , 

288     IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


r 

7! 

^o 

7! 

[)° 



/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

7212.0 

1276.9 

1643.1 

4C40.0 

7289.5 

1.30S.5 

;696.0 

4723.4 

0 

7214.6 

1278.0 

1644.8 

4642.8 

7292.1 

1309.5 

1097.7 

47  26.2 

0 

4 

7217.2 

1279.0 

1646.6 

4645.6 

7294.6 

1310.6 

1699.5 

4729.0 

4 

6 

7219.7 

1280.1 

1648.3 

4648.3 

7297.2 

1311.7 

1701.3 

4731.8 

6 

8 

7222.3 

1281.1 

1650.1 

4651.1 

7299.7 

1312.7 

1703.1 

47:34.7 

8 

10 

7224.9 

1282.2 

1651.8 

46.53.9 

7302.3 

1313.8 

1704.9 

47.37.5 

10 

1-2 

7227.5 

1283.2 

1653.6 

4656.7 

7304.9 

1314.9 

1706.6 

4740.3 

12 

14 

7230.1 

1284.3 

1655.3 

4659.4 

7307.4 

1315.9 

1708.4 

4713.1 

14 

16 

7232.7 

1285.3 

1657.1 

4662.2 

7310.0 

1317.0 

1710.2 

4745.9 

16 

18 

7235.2 

1286.4 

1658.8 

4665.0 

7312.6 

1318.1 

1712.0 

4748.7 

18 

20 

7237.8 

1287.4 

1660.6 

4667.7 

7315.1 

1.319.1 

1713.8 

4751.5 

20 

22 

7240.4 

1288.5 

1662.3 

4670.5 

7317.7 

1320.2 

1715.6 

4751.3 

22 

24 

7243.0 

1289.5 

1664.1 

4673.3 

7320.3 

1.321.3 

1717.4 

4757.1 

24 

26 

7245.6 

1290.6 

1665.8 

4676.0 

7322.8 

1322.3 

1719.2 

4760.0 

26 

28 

7248.2 

1291.6 

1667.6 

4678.8 

7325.4 

1323.4 

1721.0 

4762.8 

28 

30 

7250.7 

1292.7 

1669.3 

4681.6 

7327.9 

1324.5 

1722.8 

4765.6 

30 

32 

7253.3 

1293.7 

1671.1 

4684.4 

7330.5 

1325.5 

1724.6 

4768.4 

32 

34 

7255.9 

1294.8 

1672.8 

4687.2 

7.3.33.1 

1.3-26.6 

1726. 

4771.2 

34 

36 

"258.5 

1295.8 

1674.6 

4689.9 

7.335.6 

1327.7 

1728.2 

4774.1 

36 

38 

7261.1 

1296.9 

1676.3 

4692.7 

7338.2 

1328.7 

17:30.0 

4776.9 

38 

40 

7263.7 

1297.9 

1678.2 

4695.5 

7.340.8 

1.329.8 

17:31.9 

4779.7 

40 

42 

7266.2 

1299.0 

1679.9 

4698.3 

7343.3 

1330.8 

1733.7 

4782.6 

42 

44 

7268.8 

1300.0 

1681.7 

4701.1 

7345.9 

1331.9 

17:35.5 

4785.4 

44 

46 

7271.4 

1301.1 

1683.5 

4703.9 

7348.4 

13:3:3.0 

1737.3 

4788.2 

46 

48 

7274.0 

1302.1 

1685.3 

4706.7 

7351.0 

1334.1 

1739.1 

4791.0 

48 

50 

7276.6 

1303.2 

1087.1 

4709.5 

7353.6 

1:3:35.2 

1740.9 

4793.9 

50 

52 

7279.2 

1304.2 

1688.8 

4712.2 

7356.1 

13:36.2 

1742.7 

4796.7 

52 

54 

7281.7 

1305.3 

1690.6 

4715.0 

7358.7 

1337.3 

1744.5 

4799.5 

54 

56 

7284.3 

1306.3 

1692.4 

4717.8 

7361.3 

13:38.4 

1746.3 

.4802.4 

56 

58 

7286.9 

1307.4 

1694.2 

4720.6 

7363.8 

1339.5 

1748.1 

4805.2 

58 

60 

7289.5 

1308.5 

1696.0 

4723.4 

7366.4 

1310.6 

17.50.0 

4808.0 

60 

/ 

80°                          1 

81° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

7366.4 

1340.6 

1750  0 

4808.0 

7442.7 

1372.8 

1805.5 

4893.9 

0 

2 

7368  9 

1341.7 

1751.8 

4810.9 

7445  2 

1373.9 

1807.3 

4896.8 

2 

4 

7371.5 

1342.7 

1753.7 

4813.7 

7447.7 

1375.0 

1809.2 

4899.7 

4 

6 

7374.0 

1343.8 

1755.5 

4816.6 

7450.3 

1:376.1 

1811.1 

4902.6 

6 

8 

7376.6 

1344.9 

1757.4 

4819.4 

7452.8 

1.377.1 

1813.0 

4905.4 

8 

10 

7379.1 

1346.0 

1759.2 

4822.3 

7455.3 

1:378.2 

1814.9 

4908.3 

10 

12 

7381.7 

1347.0 

1761.0 

4825.1 

74.57.8 

1:379.3 

1816.8 

4911.2 

12 

14 

7:384.2 

1348.1 

1762.9 

4828.0 

7460.4 

1:380.4 

1818.6 

4914.1 

14 

16 

7386.7 

1349.2 

1764.7 

4830.8 

7462.9 

1381.4 

1820.5 

4917.0 

16 

18 

7389.3 

1350.3 

1766.6 

4833.7 

7465.4 

1:382.5 

1822.4 

4919.9 

18 

20 

7391.8 

1351.3 

1768.4 

4836.5 

7467.9 

1:383.6 

1824.2 

4922.8 

20 

22 

7394.4 

1352.4 

1770.2 

4839.4 

7470.4 

1:384.7 

1826.1 

4925.7 

24 

7396.9 

1353.5 

1772.1 

4842.2 

7473.0 

1.385.7 

1828.0 

4928.6 

24 

26 

7399.5 

13.54.6 

1773.9 

4845.1 

7475.5 

1:386.8 

1829.9 

4931.5 

26 

28 

7402.0 

1.355.6 

1775.8 

4847.9 

7478.0 

i;387.9 

1831.8 

49:34.4 

28 

30 

7404.5 

1356.7 

1777.6 

4850.8 

7480.5 

1:389.0 

18:33.7 

49:37.2 

30 

32 

7407.1 

1357.8 

1779.4 

48.53.7 

7483.1 

1390.1 

18:35.6 

4940.2 

32 

34 

7409.6 

1358.9 

1781.3 

48.56.5 

7485.6 

1391.2 

18:37.5 

4943.1 

:34 

36 

7412.2 

1359.9 

1783.1 

4859.4 

7488.1 

1:392.3 

18:39.4 

4946.0 

36 

38 

7414.7 

1361.0 

1785.0 

4862.3 

7490.6 

1393.4 

1841.3 

4948.9 

38 

40 

7417.3 

1362.1 

1786.8 

4865.1 

7493.2 

1394.5 

1843.2 

4951.8 

40 

42 

7419  8 

1363.2 

1788.6 

4868.0 

7495.7 

1:395.6 

1845.1 

4954.7 

42 

44 

7422.3 

1364.2 

1790.5 

4870.9 

7498.2 

1396.7 

1847.0 

49.57.6 

44 

46 

7424.9 

1365.3 

1792.4 

4873.8 

7500.7 

1397.8 

1848.9 

4960.6 

46 

48 

7427.4 

1366.4 

1794  3 

4876.6 

7503.3 

1:398.9 

18.50.8 

4963.5 

48 

50 

7430.0 

1367.5 

1796.2 

4879.5 

7505.8 

1400.0 

1852  7 

4966.4 

50 

52 

7432.5 

1368.5 

1798.0 

4882.4 

7.508.3 

1401.1 

1854.6 

4969.3 

52 

54 

7435.1 

1369.6 

1799.9 

4885  3 

7510.8 

1402.2 

1856.5 

4972.2 

54 

56 

7437.6 

1370  7 

1801.8 

4888. 1 

7513.3 

1403.3 

1858.4 

*4975.1 

56 

58 

7440.1 

1371. P 

1803.7 

4891.0 

7515.9 

1404  4 

1860.3 

4978.0 

58 

60 

1  7442.7 

1372.8 

1805.5 

4893.9 

7518.4 

1405.5 

1862  3 

4981.0 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.      289 


/ 

82° 

83° 

/ 

L.  C. 

7518.4 

1405.5- 

E. 

1862.3 

T 

4981.0 

L.  C. 

M. 

E. 

T. 

0 

7593.6 

1438.5 

1920.6 

5069.4 

0 

2 

7520.9 

1406.6 

1864.2 

4983.9 

7596.1 

1439.6 

1922.6 

5072.4 

2 

4 

T523.4 

1407.7 

1866.1 

4986 . 8 

7598.6 

1440.7 

1924.6 

5075.4 

4 

6 

7525.9 

1408  8 

1^8.1 

49S9.8 

7601.1 

1441.8 

1926.5 

5078.4 

6 

8 

7528.4 

1409.9 

1870.0 

4992.7 

7603.6 

1442.9 

1928.5 

5081.4 

8 

10 

7530.9 

1411.0 

1871.9 

4995.7 

7606.0 

1444.0 

1930,5 

5084.4 

10 

12 

7533.4 

1412.1 

1873  9 

4998.6 

7608.5 

1445.1 

1932.4 

5087.3 

12 

14 

7535.9 

1413.2 

1875.8 

5001.5 

7611.0 

1446.2 

1934.4 

5090.3 

14 

16 

7538.5 

1414.3 

1877.7 

5004.5 

7613.5 

1447.3 

1936.4 

5093,3 

16 

18 

7541.0 

1415.4 

1879.7 

5007.4 

7616.0 

1448.4 

1938.4 

5096.3 

18 

20 

7543.5 

1416.5 

1881.6 

5010.3 

7618.5 

1449.6 

1940.4 

5099.3 

20 

22 

7546.0 

1417.6 

1883.5 

5013.3 

7621.0 

1450.7 

1942.4 

5102.3 

22 

24 

7548.5 

1418.7 

1885.5 

5016.2 

7623.5 

1451.8 

1944.4 

5105.2 

24 

26 

7551.0 

1419.8 

1887.4 

5019.2 

7626.0 

1452.9 

1946.4 

5108.2 

26 

28 

7553.5 

1420.9 

1889.3 

5022.1 

7628.5 

1454.0 

1948.4 

5111.2 

28 

30 

7556.0 

1422.0 

1891. 3 

5025.0 

7030.9 

1455.1 

1950.4 

5114  2 

30 

32 

7558.5 

1423.1 

1893.2 

5028.0 

7633.4 

1456.2 

1952.4 

5117.2 

32 

34 

7561.0 

1424.2 

1895.1 

5031 .0 

7G35.9 

1457.3 

1954.4 

5120.2 

34 

36 

7563.5 

1425.3 

1897.1 

5033.9 

7038.4 

1458.4 

1956.4 

5123.2 

36 

38 

7566.0 

1426.4 

1899.0 

5036.9 

7640.9 

1459.5 

1958.4 

5126.2 

38 

40 

7568.5 

1427.5 

1901.0 

5039.8 

7643.4 

1460.7 

1960.4 

5129.2 

40 

42 

7571.0 

1428.6 

1902.9 

5042.8 

7645.9 

1461.8 

1962.4 

5132.2 

42 

44 

7573.5 

1429.7 

1904.9 

5045.8 

7648.4 

1462.9 

1964.4 

5135.2 

44 

46 

7576.1 

1430.8 

1906.9 

5048.7 

7650.9 

1464.0 

1966.4 

5138.2 

46 

48 

7578.6 

1431.9 

1908.8 

5051.7 

7653.4 

1465.1 

1968.4 

5141.2 

48 

50 

7581.1 

1433.0 

1910.8 

.50.54.6 

7655.8 

1466.2 

1970.4 

5144.3 

50 

52 

7583.6 

1434.1 

1912.8 

5057.6 

7658.3 

1467.3 

1972.4 

5147.3 

52 

51 

7586.1 

1435.2 

1914.7 

.5060.6 

7660.8 

1468.4 

1974.4 

5150.3 

54 

56 

7588.6 

1436.3 

1916.7 

.5063.5 

7663.3 

1469.5 

1976.4 

5153.3 

56 

58 

7591 . 1 

1437.4 

1918.7 

5066.5 

7665.8 

1470.6 

1978.4 

5156.3 

58 

60 

7593.6 

1438.5 

19:>0.6 

.5069.4 

7068.3 

1471.8 

1980.5 

5159.3 

60 

/ 

84°                           1 

85° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

1 

0 

7668.3 

1471.8 

1980.5 

5159.3 

7742.4 

1505.4 

2041.8 

5250.6 

0 

2 

7670.8 

1472.9 

1982.5 

5162.3 

7744.8 

1506.5 

2043.9 

5253.6 

'2 

4 

7673.2 

1474.0 

1984.5 

5165.3 

7747.3 

1507.6 

2046.0 

5256.7 

4 

6 

7675.7 

1475.1 

1986.6 

5168.4 

7749.7 

1508.8 

2048.0 

5259.8 

6 

8 

7678.2 

1476.2 

1988.6 

5171.4 

7752.2 

1509.9 

2050.1 

5262.9 

8 

10 

7680.6 

1477.4 

1990.6 

5174.4 

7754.6 

1511.0 

2052.2 

5266  0 

10 

12 

7683.1 

1478.5 

1992.7 

5177.5 

77.57.1 

1512.2 

2054.2 

5269.0 

12 

14 

7685  6 

1479.6 

1994.7 

5180.5 

7759  5 

1513.3 

20.56.3 

5272.1 

14 

16 

7688.1 

1480.7 

1990.7 

5183.5 

7762.0 

1514.4 

2058.4 

5275.2 

16 

18 

7690.5 

1481.8 

1998.8 

5186.6 

7764  4 

1515.6 

2060.5 

5278.3 

18 

20 

7693.0 

1483.0 

2000.8 

5189.6 

7766.9 

1.516.7 

2062.6 

5281.4 

20 

22 

7695.5 

1484  1 

2002.8 

5192.6 

7769.3 

1517.8 

2064.7 

5284.4 

22 

24 

7697.9 

148.T.2 

2004.9 

5195.6 

7771.8 

1519.0 

2066.8 

5287.5 

24 

26 

7700.4 

1486.3 

2006.9 

5198.7 

7774.2 

1520.1 

2068.9 

5290.6 

26 

28 

7702.9 

1487  4 

2008.9 

5201.7 

4 1 iV. 1 

1521.2 

2071.0 

5293.7 

28 

30 

7705.3 

1488  6 

2011.0 

5204.7 

7779.1 

1522.4 

2073.1 

5296.7 

30 

32 

7707.8 

1489.7 

2013.0 

5207.8 

7781.5 

1523.5 

2075.2 

5299.8 

32 

34 

7710.3 

1490.8 

2015.0 

5210  8 

7784.0 

1524.6 

2077  3 

5302.9 

34 

36 

7712.8 

1491.9 

2017.0 

5213.9 

7786.4 

1525.8 

2079.4 

5306.1 

36 

38 

7715.2 

1493.0 

2019.1 

5216.9 

7788.9 

1526.9 

2081.5 

5309.2 

38 

40 

7717.7 

1494.2 

2021.2 

5220.0 

7791.3 

1528.0 

2083.7 

.5312.3 

40 

42 

7720.2 

1495.3 

2023.2 

.5223.1 

7793.8 

1529.2 

2085.8 

5315.4 

42 

44 

7722  6 

1496.4 

2025.3 

5226.1 

7796.2 

1.530.3 

2087.9 

5318.5 

44 

46 

7725.1 

1497.5 

2027.4 

.5229.2 

7798  7 

1531 .4 

2090.0 

.5321.6 

46 

48 

7727.6 

1108.6 

2029.4 

.5232.2 

7801 . 1 

1532.6 

2092.1 

.5324.7 

48 

50 

7730.0 

1499  8 

2031.5 

5235.3 

7803.6 

1.533.7 

2094.2 

.5327.8 

50 

52 

7732  5 

1.500.9 

2033.0 

5238.3 

7806.0 

15.34  8 

2096.3 

5330  9 

52 

54 

7735.0 

1502.0 

2035.6 

5241  4 

7808.5 

1536.0 

2098.4 

5334.0 

54 

56 

7737  5 

1503.1 

2037  7 

.5244  5 

7810.9 

1.5,37.1 

2100.6 

.5337.1 

56 

58 

7739  9 

1504.2 

2039.8 

5247.5 

7813  4 

1538.2 

2102.7 

5340.2 

58 

60 

7742.4 

1505  4 

2041.8 

.5250.6 

7815  8 

1539.3 

2104.8 

.5343.3  1 

60 
i 

290      IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


r 

/ 

86°           1 

8 

4° 

/ 

L.  C. 

M. 

E. 

.T. 

L.  C. 

31. 

1.573.6 

E. 

2169.5 

T. 

54.37.5 

0 

r815.8 

15.39.3 

2104.8 

5.343.3 

7888.5 

0 

2 

7818.2 

1540.4 

2106.9 

5.346.4 

7890.9 

1574.8 

2171.6 

5440.7 

2 

4 

7820.6 

1541.6 

2109.1 

5349.5 

7893.3 

1575.9 

2173.8 

5443.9 

4 

6 

7823.1 

1542.7 

2111.2 

.5.3.52.7 

7895.7 

1577.1* 

2176.0 

5447.1 

6 

8 

7825.5 

1543.9 

2113.4 

5355.8 

7898.1 

1578.2 

2178.2 

5450.3 

8 

10 

7827.9 

1545.0 

2115,5 

5358.9 

7900.5 

1579.4 

2180.4 

54.53.4 

10 

12 

7830.3 

1546.1 

2117.6 

5362.0 

7903.0 

1580.5 

2182.5 

5456.6 

12 

14 

78.32.8 

1547.3 

2119.8 

5365.2 

7905.4 

1581.7 

2184.7 

5459.8 

14 

16 

78.35.2 

1548.4 

2121.9 

5368.3 

7907.8 

1.582.9 

2186.9 

5463.0 

16 

18 

7837.6 

1549.6 

2124.1 

5371.4 

7910.2 

1584.0 

2189.1 

5466.2 

18 

20 

7840.0 

15.50.7 

2126.2 

5374.6 

7912.6 

1.585.1 

2191.3 

5469.4 

20 

22 

7812.4 

1551.8 

2128.3 

5377.7 

7915.0 

1.586.3 

2193.5 

5472.5 

22 

24 

7844.9 

1.553.0 

21.30.5 

5?80.8 

7917.4 

1587.4 

2195.7 

5475.7 

24 

26 

7847.3 

1554.1 

2132.6 

•5383.9 

7919.8 

1588.6 

2197.9 

5478.9 

26 

28 

7849.7 

1555.3 

2134.8 

5.387.1 

7922.2 

1589.7 

2200.1 

5482.1 

28 

30 

7852.1 

15.56.4 

2136.9 

.5390.2 

7924.6 

1590.9 

2202.3 

5485.3 

30 

32 

7854.6 

1557.5 

2139.0 

5393.4 

7927.1 

1592.0 

2204.5 

5488.5 

32 

34 

7857.0 

1.5.58.7 

2141.2 

5.396.5 

7929.5 

1593.2 

2206.8 

5491.7 

34 

36 

7859.4 

1559.8 

2143.3 

5399.7 

79.31.9 

1594.3 

2209.0 

5494.9 

36 

38 

7861.8 

1561.0 

2145.5 

5402.8 

7934.3 

1595.5 

2211.2 

5498.1 

38 

40 

7864.3 

1.562.1 

2147.7 

5406.0 

7936.7 

1596.6 

2213.4 

5501.3 

40 

42 

7866.7 

1563.2 

2149.8 

5409.1 

7939.1 

1597.8 

2215.6 

5504.5 

42 

44 

7869.1 

1.564.4 

21.52.0 

5412.3 

7941.5 

1.598.9 

2217.8 

5507.7 

44 

46 

7871.5 

1.565.5 

21.54.2 

.5415.4 

7913.9 

1600.1 

2220.0 

5510.9 

46 

48 

7874.0 

1566.7 

21.56.4 

.5418.6 

7946.3 

1601.2 

22-22.3 

5514.1 

48 

50 

7876.4 

1.567.8 

2158.6 

5121.8 

7948.7 

1602.4 

2224.5 

5517.3 

50 

52 

7878.8 

1568.9 

2160.7 

5424.9 

7951.2 

1603.5 

2226.7 

5520.5 

52 

54 

7881.2 

1570.1 

2162.9 

5428.1 

79.53.6 

1604.7 

2228.9 

5523.7 

54 

56 

7883.6 

1.5:1.2 

21C5.1 

5431.2 

7956.0 

1605.8 

2231.1 

5526.9 

56 

58 

7886.1 

1.572.4 

2167.3 

54.34.4 

7958.4 

1607.0 

2233.3 

.5530.1 

58 

60 

7888.5 

1573.6 

2169.5 

5437.5 

7960.8 

1608.2 

2235.6 

55.33.3 

60 

> 

/ 

88°           1 

89° 

f 

L.  C. 

7960.8 

M. 

1608.2 

E. 
2235.6 

T. 

.5,533.3 

L.  C. 

M. 

E. 

T. 

0 

8032.4 

1643.0 

2303.6 

5630.8 

0 

o 

7963.2 

1609.4 

2237.8 

5536.6 

8034.8 

1644.1 

2305.9 

56.34.1 

2 

4 

7965.6 

1610.5 

2240.1 

.5539.8 

80.37.1 

1645.3 

2308.2 

56.37.4 

4 

6 

7968.0 

1611.7 

2-242.3 

.5543.1 

8039.5 

1646.5 

2310.5 

.5640.7 

6 

8 

7970.3 

1612.8 

2244.6 

5546.3 

8041.9 

1647.7 

2312.8 

5644.0 

8 

10 

7972.7 

1614.0 

2246.8 

5549.5 

8044.2 

1648.9 

2315.1 

5647.3 

10 

1? 

7975.1 

1615.2 

2249.1 

55.52.8 

8046.6 

1650.0 

2317.4 

5650.6 

12 

14 

7977.5 

1616.3 

2251.3 

5556.0 

8049.0 

1651.2 

2.319.7 

56.53.9 

14 

16 

7979.9 

1617.5 

2253.6 

5.559.2 

8051.4 

1652.4 

2.322.0 

.56.57.1 

16 

Ih 

7982.3 

1618.6 

2255.8 

5562 . 5 

8053.7 

1653.6 

2.324.3 

5660.4 

18 

20 

7984.7 

1619.8 

22.58.1 

.5.565.7 

8056.1 

16.54.8 

2.326.7 

5663.7 

20 

22 

7987.1 

1621.0 

2260.4 

5568.9 

8058.5 

1655.9 

2.329.0 

5667.0 

22 

24 

7989.4 

1622.1 

2262.7 

.5.572.2 

8060.8 

16.57.1 

2.331.3 

5670.3 

24 

26 

7991.8 

1623.3 

2264.9 

5.575.4 

8063.2 

1658.3 

2333.7 

5673.6 

26 

28 

7994.2 

1624.4 

2267.2 

.5.578.6 

8065.6 

1659.5 

2.3.36.0 

5676.9 

28 

30 

7996.6 

1625.6 

2269.5 

.5581.9 

8067. 9 

1660.7 

2338.3 

.5680.2 

30 

32 

7999.0 

1626.8 

2271.7 

5585.1 

8070.3 

1661.8 

2.340.7 

5683.5 

32 

34 

8001.4 

1627.9 

2273.9 

.5588.4 

8073.7 

1663.0 

2343.0 

.5686.8 

34 

36 

8003.8 

1629.1 

2276.2 

5.591.7 

8075.1 

1664.2 

2.345.3 

5690.2 

36 

38 

8006.1 

16.30.2 

2278.5 

5594.9 

8077.4 

1665.4 

2347.7 

5693.5 

38 

40 

8008.5 

1631.4 

2280.8 

.5598.2 

8079.8 

1606.6 

2.350.0 

5696.8 

40 

42 

8010.9 

1632.6 

2283.0 

.5601.4 

80S2.2 

1667.7 

2352.3 

5700.1 

42 

44 

8013.3 

1633.7 

2285.3 

.5604.7 

8084.5 

1668.8 

2.354.7 

5703.4 

44 

46 

8015.7 

10:J4.9 

2J87.6 

5608.0 

8086.9 

1670.0 

2:357.0 

5706.8 

46 

48 

8018.1 

1636.0 

2289  9 

.5611.2 

80S9.3 

1671.2 

2359.3 

.5710.1 

48 

50 

8020.5 

1637.2 

2292  2 

.5614  5 

8091.6 

1672.4 

2.361.7 

5713.4 

50 

52 

8022.9 

16:38.4 

2294! 4 

5617.8 

8094.0 

1673.5 

2.364.0 

5716.7 

52 

54 

8025.2 

1639.5 

2296.7 

.5621.0 

8096.4 

1674  7 

2366.3 

5720.0 

54 

56 

8027.6 

1640.7 

2290.0 

.562 ».  3 

8098.8 

1675.9 

2368.7 

5723.4 

56 

58 

8030.0 

1641.8 

2301 .3 

.5627.5 

8101.1 

1677.1 

2371.0 

5726.7 

58 

60 

80.32.4 

1643.0 

2303.6 

5630.8 

8103.5 

1678.3 

2373.4 

.5730.0 

60 

IX.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.      291 


/ 

yo^                  1 

91° 

1 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

8103.5 

1678.3 

2373.4 

5730.0 

8173.9 

1713.8 

2445.1 

5830.9 

0 

2 

8105.8 

1679.5 

2375.8 

5733.3 

8176.2 

1715.0 

2447.5 

5834.3 

2 

4 

8108.2 

1680.6 

2378.2 

5736.7 

8178.5 

1716.2 

2450.0 

5837.7 

4 

6 

8110.5 

1681.8 

2380.5 

.5740.0 

8180.9 

1717.4 

2452.4 

5841.1 

6 

8 

8112.9 

1683.0 

2382.9 

5743.4 

8183.2 

1718.6 

2454.8 

5844.5 

8 

10 

8115.2 

1684.2 

2385.3 

.5746.7 

81S5.5 

1719.7 

2457.2 

5847.9 

10 

12 

8117  G 

1685.4 

2387.6 

.5750.0 

8187.9 

1720.9 

2459.7 

5851.3 

12 

14 

8119.9 

1686.5 

2390.0 

5753.4 

8190.2 

1722.1 

2462.1 

5854.7 

14 

16 

8122.3 

1687.7 

2392.4 

5756.7 

8192.5 

1723.3 

2464.5 

5858.1 

16 

18 

8124.6 

1688.9 

2394.7 

5760.1 

8194.8 

1724.5 

2467.0 

5861.5 

18 

20 

8127.0 

1690.1 

2397.1 

5763.4 

8197.2 

1725.7 

2469.4 

5864.9 

20 

22 

8129.3 

1691.3 

2399.5 

.5766.8 

8199.5 

1726.9 

2471.9 

5868.3 

22 

24 

8131.7 

1692.5 

2401.9 

.5770.1 

8201.8 

1728.1 

2474.3 

5871.8 

24 

26 

8134.0 

1693.6 

2404.3 

5773.5 

8204.2- 

1729.3 

2476.7 

5875.2 

26 

28 

8136.4 

1694.8 

2106.6 

.5776.9 

8206.5 

1730.5 

2479.2 

5878.6 

28 

30 

8138.7 

1696.0 

2409.0 

5780.2 

8208.8 

1731.7 

2481.6 

5882.0 

30 

32 

8141.1 

1697.2 

2411.4 

.5783.6 

8211.1 

1732.9 

2484.1 

5885.4 

32 

34 

8143.4 

169S.4 

2413.8 

5787.0 

8213.5 

1734.1 

2486.5 

5888.9 

34 

36 

8145.8 

1699.6 

2416.2 

5790.3 

8215.8 

1735.3 

2489.0 

.5892.3 

36 

38 

8148.1 

1700.7 

2418.6 

5793.7 

8218.1 

1736.4 

2491.5 

5895.7 

38 

40 

8150.4 

1701.9 

2421.0 

5797.1 

8220.4 

1737.6 

2493.9 

5899.2 

40 

42 

8152.8 

1703.1 

2423.4 

5800.4 

8222.8 

1738.8 

2496.4 

5902.6 

42 

44 

8155.1 

1704.3 

2425.8 

.5803.8 

8225.1 

1740.0 

2498.9 

5906.0 

44 

46 

8157.5 

1705.5 

2428.2 

5807.2 

8227.4 

1741.2 

2501.3 

5909.4 

46 

48 

8159.8 

1706.7 

2430.6 

5810.6 

8229.7 

1742.4 

2.503.8 

5912.9 

48 

50 

8162.2 

1707.9 

2433.0 

.5814.0 

8232.0 

1743.6 

2506.3 

5916.3 

50 

52 

8164.5 

1709.0 

2435.4 

5817.3 

8234.3 

1744.8 

2.508.7 

5919.8 

52 

54 

8166.8 

1710.2 

2437.9 

.5820.7 

8236.7 

1746.0 

2511.2 

5923.2 

54 

56 

8169.2 

1711.4 

2440.3 

5824 . 1 

8239.0 

1747.2 

2513.7 

5926.7 

56 

58 

8171.5 

1712.6 

2442.7 

5827.5 

8241.3 

1748.4 

2516.2 

5930.1 

58 

60 

8173.9 

1713.8 

2445.1 

5830.9 

8243.6 

1749.6 

2518.7 

5933.6 

60 

0 

92°           1 

93° 

/ 

L.  C 

8243.6 

M. 
1749.6 

E. 

2518.7 

T. 

.5933.6 

L.  C. 

M. 

E. 

T. 

8312.8 

1785.7 

2594.2 

6038.2 

0 

2 

8245.9 

1750.8 

2521.2 

5937.0 

8315.1 

1786.9 

2596.8 

6041.7 

2 

4 

8248.2 

1752.0 

2.523.6 

5940.5 

8317.4 

1<88.2 

2599.3 

6045.2 

4 

6 

8250.6 

1753.2 

2.526.1 

5944.0 

8319.7 

1789.4 

2601.9 

6048.7 

6 

\ 

8252.9 

1754.4 

2528.6 

5947.4 

8322.0 

1790,6 

2604.4 

6052.2 

8 

10 

8255.2 

1755.6 

2531.1 

59.50.9 

8324.3 

1791.8 

2607.0 

6055.8 

10 

12 

8257.5 

1756.8 

2.533.6 

5954.4 

8326.6 

1793.0 

2009.6 

6059.3 

12 

14 

8259.8 

1758.0 

2536.1 

59.57.8 

8328.8 

1794.2 

2612.1 

6062.8 

14 

16 

8262.2 

1759.2 

2538.6 

5961.3 

8331.1 

1795.4 

2614.7 

6066.4 

16 

18 

8264.5 

1760.4 

2541.1 

5964.8 

8:«3.3 

1796.6 

2617.3 

6069.9 

18 

20 

8266.8 

1761.6 

2543.6 

5968.2 

8335.6 

1797.8 

2619.8 

6073.4 

20 

22 

8269.1 

1762.8 

2546.1 

5971.7 

8337.9 

1799.1 

2622.4 

6077.0 

22 

24 

8271.4 

1764.0 

2.548.6 

5975.2 

8340.2 

1800.3 

2625.0 

6080.5 

24 

26 

8273.7 

1765.2 

2551.2 

5978.7 

8342.5 

1801.5 

2627.6 

6084.1 

26 

28 

8276.0 

1766.4 

2553.7 

5982.2 

8344.8 

1802.7 

2630.2 

6087.6 

28 

30 

8278.3 

1767.6 

2.556.2 

5985.6 

8347.1 

1803.9 

2632.7 

6091.2 

30 

32 

8280.6 

1768.8 

2558.7 

5989.1 

8349.4 

1805.1 

2635.3 

6094  7 

32 

34 

8282.9 

17^0.0 

2561.2 

5992.6 

8351.7 

1806.3 

2637.9 

6098.3 

34 

36 

8285.2 

1771.2 

2563.8 

5996.1 

83.54.0 

1807.6 

2640.5 

6101.8 

36 

38 

8287.5 

1772.5 

2566.3 

5999.6 

8356.3 

1808.8 

2643.1 

6105.4 

38 

40 

8289.8 

1773.7 

2.568.8 

6003.1 

8358.5 

1810.0 

2645.7 

6109.0 

40 

42 

8292.1 

1774.9 

2.571.3 

6006.6 

8360.8 

1811.2 

2648.3 

6112.5 

42 

44 

8294.4 

1776.1 

2573.9 

6010.1 

8363.1 

1812.4 

2650.9 

6116.1 

44 

46 

8296.7 

li  1 1 .6 

2576.4 

6013.0 

8365.4 

1813.6 

2653.5 

6119.7 

46 

48 

8299.0 

1778.5 

2578.9 

6017.1 

8367.7 

1814.9 

2656.1 

6123.2 

48 

50 

8301.3 

1779.7 

2581.5 

6020.6 

8369.9 

1816.1 

2658.7 

6126.8 

50 

52 

8303.6 

1780.9 

2584.0 

6024.1 

8372.2 

1817.3 

2661.3 

6130.4 

52 

54 

8305.9 

1782.1 

2.586.6 

6027.6 

8374.5 

1818.5 

2663.9 

6133.9 

54 

56 

8308.2 

1783.3 

2589.1 

6031.1 

8376  8 

1819.7 

2666.6 

6137.5 

5(i 

58 

8310.5 

1784.5 

2591.7 

6034.6 

8379.1 

1820.9 

2669.2 

6141.1 

.5.S 

60 

8312.8 

1785.7 

2.591.2 

()n.3,S.2 

8381  3 

1822.2 

2671.8 

6114  7 

60 

'4\)'4 

IX.— 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 

1 

94°          1 

95° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M.     E. 

T. 

0 

8381.3 

1822.2 

2671.8 

6144.7 

8449.2 

1858.9  2751.5 

6253.2 

0 

2 

8383.6 

1823.4 

2674.4 

6148.3 

8451.5 

1860.1  2754.2 

62.56.9 

2 

4 

8385.9 

1824.6 

2677.0 

6151.9 

8453.7 

1861.3  2756.9 

6260.5 

4 

6 

8388.1 

1825.8 

2679.7 

6155.4 

8456.0 

1862.6  2759.6 

6264.2 

6 

8 

8;B90.4 

1827.0 

2682.3 

6159.0 

8458.2 

1863.8  2762.3 

6267.8 

8 

10 

8392.7 

1828.3 

2684.9 

6162.6 

8460.4 

1865.0  2765.0 

6271.5 

10 

12 

8395.0 

1829.5 

2687.6 

6166.2 

8462.7 

1866.3  2767.7 

6275.2 

12 

14 

8397.2 

1830.7 

2690.2 

6169.8 

8464.9 

1867.5  2770.4 

6278.8 

14 

16 

8399.5 

1831.9 

2692.8 

6173.4 

8467.2 

1868.7  2773.1 

6282.5 

16 

18 

8401.7 

1833.1 

2695.6 

6177.0 

8469.4 

1869.9  2775.8 

6286.2 

18 

20 

8404.0 

1834.4 

2698.1 

6180.6 

8471.7 

1871.2  2778.5 

6289.8 

20 

22 

8406.3 

1835.6 

2700.8 

6184.2 

8473.9 

1872.4  2781.2 

6293.5 

22 

24 

8408.5 

1836.8 

2703.4 

6187.8 

8476.2 

1873.6  2784.0 

6297.2 

24 

26 

8410.8 

1838.0 

2706.1 

6191.5 

8478.4 

1874.9  2786.7 

6300.9 

26 

28 

8413.1 

18::i9.3 

2708.7 

6195.1 

8480.7 

1876.1  2789.4 

6304.6 

28 

30 

8415.3 

1840.5 

2711.4 

6198.7 

8482.9 

1877.3  2792.1 

6308.2 

30 

32 

8417.6 

1841.7 

2714.0 

6202.3 

84S5.1 

1878.6  2794.9 

6311.9 

32 

34 

8419.9 

1842.9 

2716.7 

6205.9 

8487.4 

1879.8  2797.6 

6315.6 

34 

36 

8422.1 

1844.2 

2719.3 

6209.5 

8489.6 

1881.0  2800.3 

6319.3 

36 

38 

8424.4 

1845.4 

2722.0 

6213.2 

8491.9 

1882.3  2803.1 

6323.0 

38 

40 

8426.6 

1846.6 

2724.7 

6216.8 

8494.1 

1883.5  2805.8 

6326.7 

40 

42 

8428.9 

1847.8 

2727.3 

6220.4 

8496.3 

1884.8  2808.6 

6330.4 

42 

44 

8431.2 

1849.1 

2730.0 

6224.1 

8498.6 

18S6.0  2811.3 

6-334.1 

44 

46 

8433.4 

1850.3 

2732.7 

6227.7 

8500.8 

1887.2  2814.1 

6.337.8 

46 

48 

8435.7 

1851.5 

2735.4 

6231.3 

8503.0 

1888.5  2816.8 

6341.5 

48 

50 

8437.9 

1852.7 

2738.0 

6235.0 

8505.3 

1889.7  2819.6 

6345.2 

50 

52 

8440.2 

1854.0 

2740.7 

6238.6 

8507.5 

1890  9  2822  3 

6349  0 

52 

54 

8442.4 

18.15.2 

2743.4 

6242.3 

8509.8 

1892.2  2825.1 

6352.7 

54 

56 

8444.7 

1856.4 

2746.1 

6245.0 

8512.0 

1893.4  2827.8 

6356.4 

56 

58 

8447.0 

1857.6 

2748.8 

6249.6 

8514.2 

1894.6  2830.6 

6360.1 

58 

60 

8449.2 

ia58.9 

2751.5 

62.53.2 

8516.4 

1895.9  2833.4 

6363.8 

60 

/ 

96°           1 

97" 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M.    E. 

T. 

0 

8516.4 

1895.  S 

2833.4 

6363.8 

8583.0 

1933.2  2917.5 

6476.6 

0 

2 

8518.7 

1897.1 

2836.1 

6367.5 

8.585.2 

1934.4  2920.3 

6480.4 

2 

4 

8520.9 

1898.4 

2838.9 

6371.3 

8587.5 

1935.7  2923.2 

6484.2 

4 

6 

8.523.1 

1899.6 

2841.7 

6375.0 

8589  7 

1936.9  2926.0 

6488.0 

6 

8 

8.525.4 

1900.8 

2844.5 

6378.7 

8591.9 

1938.2  2928.9 

6491.8 

8 

10 

8527.6 

1902.1 

2847.2 

6382.5 

8594  1 

1939.4  2931.7 

6495.6 

10 

12 

a529.8 

1903  3 

28.50.0 

6386.2 

8.596.3 

1940.7  2934.6 

6499.4 

12 

14 

8532.0 

1904.6 

2852  8 

6389.9 

8.59^.5 

1941.9  2937.5 

6503.2 

14 

16 

8534.3 

1905.8 

2855  6 

6393.7 

8600.7 

1943.2  2940  3 

6.507.1 

16 

18 

8536.5 

1907.0 

2858.4 

6397.4 

8602.9 

1944.4  2943.2 

6510.9 

18 

80 

8538.7 

1908.3 

2861.2 

6401.2 

8605.1 

1945.7  2946.1 

6514.7 

20 

22 

8.540  9 

1909.5 

2864.0 

6404.9 

8607.3 

1946.9  2948.9 

6518.5 

22 

24 

8543.2 

1910.8 

2866.7 

6408.7 

8609.5 

1948.2  2951.8 

6522.3 

24 

26 

8545.4 

1912.0 

2869.5 

6412.4 

8611.7 

1949.4  2954.7 

6526.2 

26 

28 

8547. e 

1913  3 

2872.3 

6416.2 

8613.9 

1950.7  2957.6 

6530.0 

28 

30 

8549  8 

1914  5 

2875.1 

6419.9 

8616  1 

1952.0  2960.4 

6533.8 

30 

32 

85.52.0 

1915.7 

2877.9 

642;3.7 

8618.3 

1953.2  2963.3 

6537.7 

32 

34 

8554.3 

1917.0 

2S80.8 

6427  5 

8620.5 

1954.5  2966.2 

6.541.5 

34 

36 

8556.5 

1918.2 

2883.6 

6431  2 

8622.7 

19.55.7  2969.1 

6545.3 

36 

38 

8558.7 

1919.5 

2886  4 

6435  0 

8624.9 

1957.0  2972.0 

6549.2 

38 

40 

8560.9 

1920.7 

2889.2 

6438.8 

8627.1 

19.58. 2  2974.9 

65.53.0 

40 

42 

8563.1 

1922.0 

2892.0 

6442  5 

8629.3 

1959  5  2977.8 

6.5.56.9 

42 

44 

8565  3 

1923.2 

2894  8 

6446.3 

8631.5 

1960.7  2980.7 

6.560.7 

44 

46 

8567  6 

1924  5 

2897.7 

6450.1 

8633  7 

1962.0  2983.0 

6564.6 

46 

48 

8569  8 

1925.7 

2900.5 

6453.9 

8635.8 

1963.2  2986.5 

6568.4 

48 

50 

8.572.0 

1927.0 

2903.3 

64.57.6 

8638.0 

1961.5  2989.4 

6572.3 

50 

52 

8.574.2 

1928.2 

2906.1 

6461.4 

8640.2 

1965.8  2992.3 

6.576.2 

52 

54 

8576.4 

1929  4 

2909.0 

6465.2 

8642.4 

1967.0  2995  2 

6.580.0 

54 

56 

8.578.6 

1930.7 

2911.8 

6469.0 

8644.6 

1968.3  2998.1 

6.583.9 

56 

58 

8.5S0.8 

1931.9 

2914.7 

6472.8 

8646  !< 

1969.5  3001.1 

6.587.7 

58 

,  60 

8583.0 

1933.2 

2917.5 

6476.6 

8649  0 

1970.8  3004.0 

6591.6 

60 

IX.— FUNCTIONS   OF    A.   ONE-DEGREE   CURVE.      293 


0 

98° 

99° 

.' 

L.  C. 

M. 

£. 

T. 

L.  C. 

M. 

E. 

T. 

8649.0 

1970.8 

3004.0 

6591.6 

8714.3 

2008.7 

3092.9 

6709.0 

0 

2 

8651.2 

1972.0 

3006.9 

6595.5 

8716.4 

2009.9 

3095.9 

6712.9 

2 

4 

8653.3 

1973.3 

3009.8 

6.599.4 

8718.6 

2011.2 

3098.9 

6716.9 

4 

6 

8655.5 

1974.6 

3012.8 

6603.2 

8720.7 

2012.5 

3101.9 

6720.8 

6 

8 

8657.7 

1975.8 

3015.7 

6607.1 

8722.9 

2013.7 

3104.9 

6724.8 

8 

10 

8659.9 

1977.1 

3018.6 

6611.0 

8725.1 

2015.0 

3107.9 

6728.8 

10 

12 

8662.1 

1978.3 

3021.6 

6614.9 

8727.2 

2016.3 

3111.0 

6732.7 

12 

14 

8664.3 

1979.6 

3024.5 

6618.8 

8729.4 

2017.5 

3114.0 

6736.7 

14 

16 

8666.4 

1980.9 

3027.5 

6622.7 

8731.5 

2018.8 

3117.0 

6740.7 

16 

18 

8668.6 

1982.1 

3030.4 

6026.6 

8733.7 

2020.1 

3120.0 

6744.6 

18 

20 

8670.8 

1983.4 

3033.3 

6630.5 

8735.9 

2021.4 

3123.1 

6748.6 

20 

22 

8673.0 

1984.6 

3036.3 

6634.4 

8738.0 

2022.6 

3126.1 

6752.6 

22 

24 

8675.2 

1985.9 

3039.3 

6638.3 

8710.2 

2023.9 

3129.1 

6750.6 

24 

26 

8677.3 

1987.2 

3042.2 

6642.2 

8742.3 

2025.2 

3132.2 

6760.6 

26 

28 

8679.5 

1988.4 

8045.2 

6646.1 

8744.5 

2026.4 

3135.2 

6764.6 

28 

30 

8681.7 

1989.7 

3048.1 

6650.0 

8746.6 

2027.7 

3138.3 

6768.6 

30 

32 

8683.9 

1991.0 

3051.1 

6653.9 

8748.8 

2029.0 

3141.3 

6772.6 

32 

34 

8686.0 

1992.2 

3054 . 1 

6657.8 

8750.9 

2030.3 

3144.4 

6776.6 

34 

36 

8688.2 

1993.5 

30.57.0 

6661.7 

8753.1 

2031.5 

3147.4 

6780.6 

36 

38 

8690.4 

1994.7 

3060.0 

6665.7 

8755.3 

2032.8 

3150.5 

6784.6 

38 

40 

8692.6 

1996.0 

3063.0 

6669.6 

8757.4 

2034.1 

3153.5 

6788.6 

40 

42 

8694.7 

1997.3 

30U6.0 

6673.5 

8759.5 

2035.4 

3156.6 

6792.6 

42 

44 

8696.9 

1998.5 

3068.7 

6677.4 

8761.7 

2036.6 

3159.7 

6796.6 

44 

46 

8699.1 

199;).  8 

3071.9 

6681.4 

8763.8 

2037.9 

3162.7 

6800.6 

46 

48 

8701.2 

2001.1 

3074.9 

6685.3 

8766.0 

2039.2 

3165.8 

6804.6 

48 

50 

8703.4 

2002.3 

3077.9 

6689.2 

8768.1 

2040.5 

3168.9 

6808.6 

50 

52 

8705.6 

2003.6 

3080.9 

6693.2 

8770.3 

2041.7 

3172.0 

6812.6 

52 

54 

8707.8 

2004.9 

3083.9 

6697.1 

8772.4 

2043.0 

3175.1 

6816.7 

54 

56 

8709.9 

2006.1 

3086.9 

6701.1 

8774.6 

2044.3 

3178.1 

6820.7 

56 

58 

8712.1 

2007.4 

3089.9 

6705.2 

8776.7 

2045.6 

3181.2 

6824.7 

58 

60 

^  , 

8714.3 

2008.7 

3092.9 

6709.0 

8778.9 

2046.8 

3184.3 

6828.8 

60 

/ 

0 

100°         1 

101° 

/ 

L  C. 

8778.9 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

2046.8 

3184.3 

6828.8 

8842.8 

2085.3 

3278.3 

6951.0 

0 

o 

8781.0 

2048.1 

3187.4 

6832.8 

8844.9 

2086.6 

3281.5 

6955.2 

2 

4 

8783.1 

2049.4 

3190.5 

6836.8 

8847.0 

2087.8 

3284.7 

6959.3 

4 

6 

8785.3 

20.50.7 

3193.6 

6840.9 

8849.2 

2089.1 

3287.9 

6963.4 

6 

8 

8787.4 

2051.9 

3196.7 

6S44.9 

8851.3 

2090.4 

3291.1 

6967.6 

8 

10 

8789.6 

2053.2 

3199.8 

6849.0 

8853.4 

2091.7 

3294.3 

6971.7 

10 

12 

8791.7 

2054.5 

3202.9 

6853.0 

8855.5 

2093.0 

3297.5 

6975.8 

12 

14 

8793.9 

20.55.8 

3206.0 

6857.1 

88.57.6 

2094.3 

3300.7 

6980.0 

14 

16 

8796.0 

20.57.1 

3209.1 

6861.1 

8859.8 

2095.6 

3303.9 

6984.1 

16 

18 

8798.9 

2058.3 

3212.2 

6865.2 

8861.9 

2096.9 

3307.1 

6988.2 

18 

20 

8800.3 

2059.6 

3215.4 

6869.2 

8864.0 

2098.2 

3310.3 

6992.4 

20 

22 

8802.4 

2060.9 

3218.5 

6873.3 

8866.1 

2099.4 

3313.5 

6996.6 

22 

24 

8804.5 

2062.2 

3221.6 

6877.4 

8868.2 

2100.7 

3316.7 

7000.7 

24 

26 

8806.7 

2063.5 

3224.7 

6881.4 

8870.3 

2102.0 

3319.9 

7004.9 

26 

28 

8808.8 

2064.7 

3227.9 

6885.5 

8872.4 

2103.3 

3323.1 

7009  0 

28 

30 

8810.9 

2066.0 

3231  0 

6889.6 

8874.5 

2104.6 

3326.4 

7013.2 

30 

32 

8813.1 

2067.3 

3234.1 

6893.7 

8876.7 

2105.9 

3329.6 

7017.3 

32 

34 

8815.2 

2068.6 

3237.3 

6897.8 

8878.8 

2107.2 

3332.8 

7021.5 

34 

36 

8817.3 

2069.9 

3240.4 

6901.8 

8880.9 

2108.5 

3336.0 

7025.7 

36 

38 

8819.5 

2071 . 1 

3243.5 

6905.9 

8883.0 

2109.8 

3339.3 

7029.9 

38 

40 

8821.6 

2072.4 

3246.7 

6910.0 

8885.1 

2111.1 

3342.5 

7034.0 

40 

42  ' 

8823.7 

2073.7 

3249.8 

6914.1 

8887.2 

2112.4 

3345.8 

7038.2 

42 

44 

8825.8 

2075.0 

3253.0 

6918.2 

8889.3 

2113.6 

3349.0 

7042.4 

44 

46 

8828.0 

2076.3 

3256.2 

6922.3 

8891,4 

2114.9 

3352.3 

7046.6 

46 

48 

8830.1 

2077.6 

3259.3 

6926.4 

8893.5 

2116.2 

3355.5 

7050.8 

48 

50 

8832.2 

2078.9 

3262.5 

6930.5 

8895.6 

2117.5 

3358.8 

7055  0 

50 

52 

8834 . 3 

2080.1 

3265.7 

6934.6 

8S97.7 

2118.8 

3362.0 

70.59.2 

52 

54 

8^36.4 

2081.4 

3268.8 

6938.7 

8899.8 

2120.1 

3365.5 

7063  4 

54 

56 

8K38  6 

2082.7 

3272.0 

6942.8 

8901.9 

2121.4 

3368.7 

7067.6 

56 

58 

S340  7 

20S4.0 

3275.2 

6946.9 

8904.0 

2122.7 

3372.0 

7071.8 

58 

60 

8842  8 

2085.3 

3278.3 

6951.0 

8906.1 

2124.0 

3375.1 

7076.0 

60 

294     IX.— FUNCTIONS   OF   A  ONE-DEGREE   CURVE. 


/ 

102° 

103° 

/ 

L.  C.   M.    E.     T. 

L.  C.   M.    E.     T. 

0 
2 

4 
6 
8 
10 
12 
14 
16 
18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 

8906.1  2124.0  3375.1  7076.0 

8908.2  2125.3  3378.3  7080.2 

8910.3  2126.6  ;3381.6  70.84. 4 

8912.4  2127.9  3384.9  7088.6 

8914.5  2129.2  3388.2  7092.8 

8916.6  2130.5  3391.5  7097.1 

8918.7  2131.8  3394  7  7101.3 

8920.8  2133.1  3398.0  7105.5 

8922.9  2134.4  3401.3  7109.7 
8925.0  2135.7  3404.6  7114.0 

8927.0  2137.0  3407.9  7118.2 

8929.1  2138.3  3411.2  7122.4 

8931.2  2139.6  3414.5  7126.7 

8933.3  2140.9  3417.9  7130.9 

8935.4  2142.2  3421.2  7135.2 

8937.5  2143.5  3424.5  71-39.4 

8939.6  2144  8  3427.8  7143.7 

8941.6  2146.1  3431.1  7148.0 

8943.7  2147.4  3434.5  7152.2 

8945.8  2148.7  3437.8  7156.5 

8947.9  2150  0  3441.1  7160.7 

8950.0  2151.3  3444.4  7165.0 

8952.1  2152.6  3447.8  7169.3 

8954.1  2153.9  3451.1  7173.6 

8956.2  2155.2  3454.5  7177.9 
89.58.3  21.56.5  34.57.8  7182.1 

8960.4  2157.8  3461.2  7186.4 

8962.5  2159.1  3464.5  7190.7 

8964.5  2160.4  3467.9  7195.0 

8966.6  2161.7  .3471.2  7199.3 

8968.7  2163.0  3474.6  7203.6 

8968.7  2163.0  3474.6  7203.6 

8970.8  2164.3  3478.0  7207.9 

8972.9  2165.6  3481.4  7212.2 
8974.9  2166.9  3484.7  7216.5 

8977.0  2168.2  3488.1  7220.8 

8979.1  2169.5  3491.5  7225.1 

8981.1  2170.8  3494.9  7229.5 

8983.2  2172.1  3498.3  7233.8 

8985.3  2173.4  3501.6  7238.1 
8987.3  2174.7  3505.3  7242.4 

8989  4  2176.1  3508.4  7246.8 
8991.5  2177.4  3511.8  7251.1 

8993.5  2178.7  3515.2  7255.4 

8995.6  2180.0  -3518.7  7259.8 

8997.7  2181.3  3522.1  7264.1 

8999.7  2182.6  3525.5  7268.5 

9001.8  2183.9  3528.9  7272.8 

9003.9  2185.2  35-32.3  7277.2 
9005.9  2186.5  3535.7  7281.5 
9008.0  2187.8  3539.2  7285.9 

9010.0  2189.1  3542.6  7290.3 

9012.1  2190.5  3,546.0  7294.6 

9014.2  2191.8  3549.5  7299,0 

9016.2  2193.1  3552.9  7303.4 

9018.3  2194.4  3556.3  7307.7 

9020.3  2195  7  3559.8  7312.1 

9022.4  2197.0  3.563.2  7316.5 

9024.5  2198.3  3566.7  7320.9 

9026.5  2199.6  3-570.2  7-325.3 

9028.6  2200.9  3573.6  7329.7 
90.30.6  2202.3  .3.577.1  7334.1 

0 
2 

4 

6 

8 
10 
12 
14 
16 
18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 
42 
44 
46 

48 
50 
52 
54 
56 
58 

60 

^ 

/ 

104° 

105° 

/ 

L.  C.   M.     E.    T. 

L.  C.   M.    E.    T. 

0 
2 
4 
6 

8 
10 
12 
14 
16 
18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 
.  42 
44 
46 
48 
50 
52 
54 
56 
58 
60 

9030.6  2202.3  3577.1  7334.1 
90.32.7  2203.6  3580.5  7388.5 

9034.7  2204.9  3584.0  7342.9 

9036.8  2206.2  3587.5  7347.3 

9038.8  2207.5  3591.0  7351.7 

9040.9  2208  8  3594.4  7356.1 
9042.9  2210  2  3597.9  7360.5 
9045.0  2211.5  3601.4  7.364.9 

9047.0  2212.8  3604.9  7369.4 

9049.1  2214.1  .3608.4  7373.8 

9051.1  2215.4  3611.9  7378.2 

90.53.1  2216.7  3615.4  7-382.6 

9055.2  2218.0  3618.9  7-387.1 

90.57.2  2219.4  3622.4  7391.5 

9059.3  2220.7  -3625.9  7396.0 
9001.3  2222.0  3629.4  7400.4 

9063.3  2223.3  3633.0  7404.8 

9065.4  2224.6  3636.5  7409.3 

9067.4  2226.0  3640.0  7413  8 

9069.5  2227.3  3643.5  7418.2 

9071.5  2228.6  .3647.1  7422.7 

9073.5  2229.9  3650.6  7427.1 

9075.6  2231.2  -3654.1  7431.6 
9077  6  2232.6  36.57.7  7436.1 

9079.6  2233.9  3661.2  7440  6 

9081.7  2235.2  3664.8  7445.0 
9083.7  2236.5  3668.3  7449.5 

9085.7  22-37.8  3671.9  74.54.0 

9087.8  2239.2  3675.4  74.58  5 
90S9.8  2240.5  3679.0  7463.0 
9091  8  2241.8  3682.6  7467  5 

9091.8  2241.8  3682.6  7467.5 

9093.9  2243.1  3686.1  7472.0 
9095.9  2244.4  3689.7  V476.5 
9097.9  2245.8  3693.3  7481.0 
9099.9  2247.1  3696.9  7485.5 
9102.0  2248.4  3700.4  7490.0 
9104.0  2249.7  3704.0  7494.5 
9106.0  2251.1  3707.6  7499.1 

9108.0  2252.4  3711.2  7503.6 

9110.1  2253.7  3714.8  7508.1 

9112.1  2255.0  3718.4  7512.6 
9114.1  22.56.4  3722.0  7517.2 
9116.1  2257.7  3725.6  7521.7 

9118.1  22.-9  0  3729.3  7.526.3 

9120.2  2260.3  3732.9  7530.8 
9122.2  2261.7  3736.5  75.35.3 
9124.2  2263.0  3740.1  7539  9 
9126.2  2264.3  3743.7  7544.4 
9128  2  2265.7  3747.4  7.549.0 

9130.2  2267.0  3751.0  7553.6 

91-32.3  2268.3  .3754.6  7558.1 

9134.3  2269.6  3758.3  7562.7 
9136.3  2271.0  3761.9  7.567.3 
9138.3  2272.3  3765.6  7571.8 
9140.3  2273  6  3769.2  7.576.4 
9142.3  2275.0  3772.9  7581.0 
9144.3  2276.3  3776.5  7.585.6 
9140  3  2277.6  3780.2  7.590.2 

9148.3  2278.9  3783.9  7594.8 

9150.4  22S0.3  3787.5  7.599.4 
9152.4  2281.6  .37912  7604  0 

0 
2 

4 
6 
8 
10 
12 
14 
16 
18 

20 
22 

24 
26 

28 
30 
32 
34 
36 
38 

40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
(iO 

IX.— FUNCTIONS  OF  A   ONE-DEGREE   CURVE.     295 


/ 

106°           1 

107» 

L  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

9152.4 

2281.6 

3791.2 

7604.0 

9212.2 

2321.7 

3903.1 

7743.7 

0 

2 

9154.4 

2282.9 

3794.9 

7608.6 

9214.2 

2323.0 

3906.9 

7748.4 

til 

4 

9156.4 

2284.3 

3798.6 

7613.2 

9216.2 

2324.4 

3910.7 

7753.1 

4 

C 

9158.4 

2285.6 

3802.3 

7617.8 

9218.1 

2325.7 

3914.5 

7757 . 8 

6 

8 

9160.4 

2286.9 

3805.9 

7622.4 

9220.1 

2327.0 

3918.3 

7762.5 

8 

10 

9162.4 

2288.3 

3809.6 

7627.0 

9222.1 

2328.4 

3922.1 

7767.3 

10 

12 

9164.4 

2289.0 

3813  3 

7631.7 

9224.1 

2329.7 

3925.9 

7772.0 

12 

14 

9166.4 

2290.9 

3817.0 

7636.3 

9226.1 

2331.1 

3929.7 

7776.7 

14 

16 

9168.4 

2292.3 

3820.7 

7610.9 

9228.1 

2832.4 

3933.6 

7781.5 

16 

18 

9170.4 

2293.6 

3824.4 

7645.5 

9230.0 

2333.7 

3937.4 

7786.2 

18 

20 

9172.4 

2294.9 

3828.1 

7650.2 

9232.0 

2335.1 

3941 .2 

7791.0 

20 

22 

9174.4 

2296.3 

3831.9 

7654.8 

9234.0 

2336.4 

3945.0 

7795.7 

22 

24 

9176.4 

2297.6 

3835.6 

7659.5 

9235.9 

2337.8 

3948.9 

7800.5 

24 

26 

9178.4 

2298.9 

3839.3 

7664.1 

9237.9 

2339.1 

3952.7 

7805.2 

26 

28 

9180.4 

2300.3 

3843.0 

7668.8 

9239.9 

2340.5 

3956.5 

7810.0 

28 

30 

9182.4 

2301.6 

3846.7 

7673.4 

9241.9 

2341.8 

3960.4 

7814.7 

30 

32 

9184.4 

2302.9 

3850.5 

7678.1 

9243.8 

2343.1 

3964.2 

7819.5 

32 

34 

9186.4 

2304.3 

3854.2 

7682.7 

9245.8 

2344.5 

3968.1 

7824.3 

34 

36 

9188.4 

2305.6 

3858.0 

7687.4 

9247.8 

2345.8 

3971.9 

7829.1 

36 

38 

9190.4 

2306.9 

3861.7 

7692.1 

9249.7 

2347.2 

3975.8 

7833.8 

38 

40 

9192.4 

2308.3 

3865.4 

7696.7 

9251.7 

2348.5 

3979.6 

7838.6 

40 

42 

9194.4 

2309.6 

3869.2 

7701.4 

9253.7 

2349.9 

3983.5 

7843.4 

42 

44 

9196.3 

2311.0 

3873.0 

7706.1 

9255.6 

2351  2 

3987.4 

7848.2 

44 

46 

9198.3 

2312.3 

3876.7 

7710.8 

9257.6 

2352.6 

3991.3 

7853.0 

46 

48 

9200.3 

2313.6 

3880.5 

7715.5 

9259.6 

2353.9 

3995.1 

7857.8 

48 

50 

9202.3 

2315.0 

3884.2 

7720.1 

9261.5 

2355.3 

3999.0 

7862.6 

50 

52 

9204  3 

2316.3 

3888.0 

7724.8 

9263.5 

2356.6 

4002.9 

7867.4 

52 

54 

9206.3 

2317.7 

3891.8 

7729.5 

9265.4 

2358.0 

4006.8 

7872.2 

54 

56 

9208.2 

2319.0 

3895.6 

7734.2 

9267.4 

2359.3 

4010.7 

7877,0 

56 

58 

9210.2 

2320.3 

3899.3 

7739.0 

9269.4 

2360.7 

4014.6 

7881.9 

58 

60 

9212.2 

2321.7 

3903.1 

7743.7 

9271 .3 

2362.0 

4018.5 

7886.7 

60 

/ 

108° 

109° 

-  ■  - 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

9271.3 

2362.0 

4018.5 

7886.7 

9329.8 

2402.6 

4137.4 

8033.2 

0 

2 

9273.3 

2363.3 

4022.4 

7891 .5 

9331.7 

2403.9 

4141.4 

8038.1 

o 

4 

9275.3 

2364.7 

4026.3 

7896.3 

9333.6 

2405.3 

4145.4 

8043.1 

4 

6 

9277.2 

2366.0 

4030.2 

7901.2 

9335.6 

2406.6 

4149.5 

8048.0 

6 

8 

9279.2 

2367.4 

4034.1 

7906.0 

9337.5 

2408.0 

4153.5 

8053.0 

8 

10 

9281.1 

2368.7 

4038.0 

7910.8 

9339.4 

2409.4 

4157.5 

8057.9 

10 

12 

9283.1 

2370.1 

4042.0 

7915.7 

9341.4 

2410.7 

4161.6 

8062.9 

12 

14 

9285.0 

2371.4 

4045.9 

7920.5 

9343.3 

2412.1 

4165.6 

8067.9 

14 

16 

9287.0 

2372.8 

4049.8 

7925.4 

9345.2 

2413.4 

4169.7 

8072.8 

16 

18 

9288.9 

2374.1 

4053.8 

7930.3 

9347.2 

2414.8 

4173.8 

8077.8 

18 

20 

9290.9 

2375.5 

4057.7 

7935.1 

9349.1 

2416.2 

4177.8 

8082.8 

20 

22 

9292.8 

2376.8 

4061.6 

7940.0 

9351.0 

2417.5 

4181.9 

8087.8 

22 

24 

9294.8 

2378.2 

4065.6 

7944.8 

9352.9 

2418.9 

4186.0 

8092.8 

24 

26 

9296.7 

2379.5 

4069.5 

7949.7 

9354.9 

2420.2 

4190.0 

8097.8 

26 

28 

9298.7 

2380.9 

4073.5 

7954.6 

9356.8 

2421.6 

4193.1 

8102.8 

28 

30 

9300.6 

2382.3 

4077.5 

7959.5 

9358.7 

2423.0 

4198.2 

8107.8 

30 

32 

9302.6 

2383.6 

4081.4 

7964.4 

9360.6 

2424.3 

4202.3 

8112.8 

32 

34 

9304.5 

2385.0 

4085.4 

7969.3 

9362.6 

2425.7 

4206.4 

8117.8 

34 

36 

9306.5 

2386.3 

4089.4 

7974.1 

9364.5 

2427.0 

4210.5 

8122.8 

36 

38 

9308.4 

2387.7 

4093.4 

7979.0 

9366.4 

2428.4 

4214.6 

8127.8 

38 

40 

9310.4 

2389.0 

4097.3 

7983.9 

9368.3 

2430.0 

4218.7 

8132.8 

40 

42 

9312.3 

2390.4 

4101.3 

7988.8 

9370.2 

2431.1 

4222.8 

8137.9 

42 

44 

9314.2 

2391.7 

4105.3 

7993.8 

9372.2 

2432.5 

4226.9 

8142.9 

44 

46 

9316.2 

2393.1 

4109  3 

7998.7 

9374.1 

2433.9 

4231.0 

8147.9 

46 

48 

9318.1 

2394.4 

4113.3 

8n03.6 

9376.0 

2435.2 

4235.1 

8153.0 

48 

50 

9320.1 

2395.8 

4117.3 

80U8.5 

9377.9 

2436.0 

4239  3 

8158.0 

50 

52 

9322.0 

2397.2 

4121.3 

8013.4 

9379.8 

2438.0 

4243.4 

8163.1 

52 

54 

9323  9 

2398.5 

4125.3 

8018.4 

9381.7 

24.39.3 

4247.5 

8168.1 

54 

56 

9325.9 

2399.9 

4129.3 

8023  3 

9383. 7 

2440.7 

4251.7 

8173.2 

56 

58 

9327,8 

2401.2 

4 1 33  4 

8028.2 

93S5.6 

2442.1 

42.55.8 

81 78  2 

58 

L  60 

9329.8 

2402 . 6 

4137.4 

8033 . 2 

93H7.5 

2443  4 

4260  0 

8183.3 

60 

y'^m!^m,''mm 


296 

IX.  FUNCTIONS 

OF  A 

ONE -DEGREE  CURVE. 

/ 

110°           1 

111 

1° 

0 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

0 

9387.5 

2443.4 

4260.0 

8183.3 

9444.5 

2484.5 

4386.4 

8337.2 

2 

9389.4 

2444.8 

4264.1 

8188.4 

9446.4 

2485.9 

4390.7 

8342.4 

2 

4 

9391.3 

2446.1 

4268.3 

8193.4 

9448.3 

2487.2 

4395.0 

8347.6 

4 

6 

9393  2 

2447.5 

4272.4 

8198.5 

9450.1 

2488.6 

4399.3 

8352.8 

6 

8 

9395.1 

2448.9 

4276.6 

8203.6 

94.52.0 

2490.0 

4403.6 

8:358.0 

8 

10 

9397.0 

2450.2 

4280.8 

8208.7 

9453.9 

2491 ,4 

4407.9 

8363.2 

10 

12 

9398.9 

2451.6 

4284.9 

8213.8 

9455.8 

2492.7 

4412.2 

8368.5 

12 

14 

9400.8 

2453.0 

4289.1 

8218.9 

94.57.7 

2494.1 

4416.5 

8373.7 

14 

16 

9402.7 

2454.3 

4293.3 

8224.0 

9459.6 

2495.5 

4420.8 

8378  9 

16 

18 

9404.7 

2455.7 

4297.5 

8229.1 

9461.4 

2496.9 

4425.1 

8384.1 

18 

20 

9406.6 

2457.1 

4301.7 

8234.2 

9463.3 

2498.2 

4429.5 

8389.4 

20 

22 

9408.5 

2458.4 

4305.9 

8239.3 

9465.2 

2499.6 

4433.8 

8394  6 

22 

24 

9410.4 

24.59  8 

4310.1 

8244.4 

9467.1 

2501.0 

4438.1 

8399.9 

24 

26 

9412.3 

2461.2 

4314.3 

8249.5 

9469.0 

2502.4 

4442.5 

8405.1 

26 

28 

9414.2 

2462.6 

4318.5 

8254.6 

9470.8 

2503.8 

4446.8 

8410.4 

28 

30 

9416.1 

2463.9 

4322.7 

8259.8 

9472.7 

2505.1 

4451.2 

8415.6 

30 

32 

9418.0 

2465.3 

4326.9 

8264.9 

9474.6 

2506.5 

4455.5 

8420.9 

32 

34 

9419.9 

2466.7 

4331.1 

8270.0 

9476.5 

2507.9 

4459.9 

8426.2 

34 

36 

9421.8 

2468.0 

4335,4 

8275.2 

9478.3 

2509.3 

4464.2 

8431.4 

36 

38 

9423.7 

2469.4 

4339.6 

8280.3 

9480.2 

2510.6 

4468.6 

8436.7 

38 

40 

9425.6 

2470.8 

4343.8 

8285.5 

9482.1 

2512.0 

4473.0 

8442.0 

40 

42 

9427.5 

2472.1 

4348.1 

8290.6 

9484.0 

2513.4 

4477.3 

8447.3 

42 

44 

9429.3 

2473.5 

4352.3 

8295.8 

9485.8 

2514.8 

4481.7 

84.52.6 

44 

46 

9431.2 

2474.9 

4356.6 

8300.9 

9487.7 

2516.2 

4486.1 

8457.9 

46 

48 

9433.1 

2476.3 

4360.8 

•8306.1 

9489.6 

2.517.5 

4490.5 

8463.2 

48 

50 

9435.0 

2477.6 

4365.1 

8311.3 

9491.4 

2518.9 

4494.9 

8468.5 

50 

52 

9436.9 

2479.0 

4369,3 

8316.5 

9493.3 

2520.3 

4499.3 

8473.8 

52 

54 

9438.8 

2480,4 

4373.6 

8.321.6 

9495.2 

2521.7 

4503.7 

8479.1 

54 

56 

9440.7 

2481 .7 

4377.9 

8326.8 

9497.0 

2523.1 

4508.1 

8484.4 

56 

58 

9442.6 

2483.1 

4382  2 

8332.0 

9498.9 

2524.5 

4512.5 

8489.7 

58 

60 

9444.5 

2484.5 

4386.4 

8337.2 

9500.8 

2525.8 

4516.9 

8495.1 

60 

/ 

11 

2» 

113° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C, 

M. 

E. 

T. 

0 

9500.8 

2525.8 

4.516.9 

8495.1 

9.5.56.3 

2.567.4 

4651.6 

8657.1 

0 

2 

9502.6 

2527.2 

4521.4 

8.500.4 

95.58.2 

2568.8 

4656.2 

8662.6 

2 

4 

9504.5 

2528.6 

4525.8 

8505.8 

9.560.0 

2570.2 

4660.8 

8668.0 

4 

6 

9506.4 

2530.0 

4530.2 

8511.1 

9561.8 

2571.6 

4665.4 

8673.5 

6 

8 

9508.2 

2531.4 

4534.6 

8516.4 

9.563.7 

2573.0 

4669.9 

8679.0 

8 

10 

9510.1 

2532.7 

4.539.1 

8.521.8 

9.565.5 

2574.4 

4674.5 

8684.5 

10 

12 

9511.9 

2534.1 

4543.5 

8527.1 

9567.4 

2.575.8 

4679.1 

8690.0 

12 

14 

9513.8 

2.535.5 

4548.0 

85.32.5 

9569.2 

2577.1 

4683.7 

8695.5 

14 

16 

9515.7 

2536.9 

4552.4 

8537.9 

9571.0 

2578.5 

4688.3 

8701.0 

16 

18 

9517.5 

2538.3 

4556.9 

8543.2 

9.572.9 

2579.9 

4692.9 

8706.5 

18 

20 

9519.4 

2539.7 

4.561.3 

8548.6 

9.574.7 

2581.3 

4697.5 

8712.0 

20 

22 

9521.2 

2541.0 

4.565.8 

8554.0 

9576.5 

2582.7 

4702.1 

8717.6 

22 

24 

9523.1 

2542.4 

4570.3 

8.5.59.4 

9.578.4 

2.584.1 

4706.8 

8723.1 

24 

26 

9524.9 

2543.8 

4574.8 

8.564.8 

9.580  2 

2585.5 

4711.4 

8728.6 

26 

28 

9526.8 

2545.2 

4.579.3 

8570.2 

9.582.0 

2586.9 

4716.0 

8734.2 

28 

30 

9528.6 

2546.6 

4583.7 

8575.6 

9.583.8 

2588.3 

4720.6 

8739.7 

30 

32 

9530.5 

2548.0 

4588.2 

8581.0 

9.585.7 

2589.7 

4725.3 

8745.3 

32 

34 

95.32.3 

2549.4 

4592.7 

8586.4 

9587.5 

2591.1 

4729.9 

8750.8 

34 

36 

9534.2 

2550.7 

4.597.2 

8591.8 

9.589.3 

2.592.5 

4734.6 

8756.4 

36 

38 

9536  0 

2552.1 

4601.7 

8597.2 

9591.1 

2593.9 

4739.2 

8761.9 

38 

40 

9537.9 

2553.5 

4606  2 

8602.6 

9.593.0 

2.595.3 

4743.9 

8767.5 

40 

42 

9539.7 

2554.9 

4610  8 

8608.0 

9594.8 

2596.7 

4748.5 

8773.1 

42 

44 

9541.6 

2556.3 

4615.3 

^6l3.5 

9596.6 

2.598.1 

4753.2 

8778.6 

44 

46 

9543.4 

2557.7 

4619.8 

8618.9 

9598,4 

2599.4 

4757.9 

8784.2 

46 

48 

9.545.3 

2559.1 

4624.3 

8624.3 

9600.3 

2600.8 

4762.6 

8789.8 

48 

50 

9.547.1 

2560.5 

4628.9 

8629.8 

9602.1 

2602.2 

4767.2 

8795.4 

50 

52 

9549.0 

2561.8 

4633  4 

8635.2 

9603.9 

2603.6 

4771.9 

8801,0 

52 

54 

9.550.8 

2563  2 

4638.0 

8640.7 

9605.7 

2605  0 

4776.6 

8806.6 

54 

56 

9552.6 

2564.6 

4642.5 

8646.2 

9607  5 

2606.4 

4781.3 

8812.2 

56 

58 

'.)554  5 

2566.0 

4647  1 

8651.6 

9609.4 

2607.8 

4786 . 0 

8817.8 

58 

60 

9.556  3 

2567.4 

4651.6 

86.57.1 

9611.2 

2609.2 

4790.7 

8823.4 

60 

IX.     FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 

297 

/ 

0 

114°                         1 

115° 

/ 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

9611  2 

2609.2 

4790.7 

8823,4 

9665.3 

2651.3 

4934.4 

8994.3 

0 

2 

9613.0 

2610.6 

4795.5 

8829.1 

9667.1 

2652.7 

4939.3 

9000.1 

2 

4 

9614.8 

2612.0 

4800.2 

8834.7 

9668.8 

2654 . 1 

4944.2 

9005.9 

4 

6 

9616.6 

2613.4 

4804.9 

8840.3 

9670.6 

2655.5 

4949.1 

9011.6 

6 

8 

9618.4 

2614.8 

4809.6 

8846.0 

9672.4 

2656.9 

4954.0 

9017.4 

8 

10 

9620.2 

2616.2 

4814.4 

8851.6 

9674.2 

2658.3 

4958.9 

9023.2 

10 

12 

9622.0 

2617.6 

4819.1 

8857.2 

9676.0 

2659.7 

4968.8 

9029.0 

12 

14 

9623.8 

2619.0 

4823.9 

8862.9 

9677.8 

2661.1 

4968.7 

9034.8 

14 

16 

9625.7 

2620.4 

4828.6 

8868.5 

9679.6 

2662.5 

4973.6 

9040.7 

16 

18 

962T.5 

2621. e 

4833.4 

8874.2 

9681.4 

2663.9 

4978.5 

9046.5 

18 

20 

9629.3 

2623.2 

4838.1 

8879.9 

9683.1 

2665.4 

4983.4 

9052.3 

20 

22 

9631 . 1 

2624.6 

4812.9 

8885.5 

9684.9 

2666.8 

4988.3 

9058.1 

22 

24 

9632.9 

2626.0 

4847.7 

8891.2 

9086.7 

266H.2 

4993.8 

9064.0 

24 

26 

9634.? 

2627.4 

4852.4 

8896.9 

9688.5 

2669.6 

4998.2 

9069.8 

26 

28 

9636.5 

2628.8 

4857.2 

8902.6 

9690.3 

2671.0 

5003.2 

9075.7 

28 

30 

9688.3 

2630.2 

4862.0 

8908.3 

9692.0 

2672.4 

5008.1 

9081.5 

30 

32 

9640.1 

2631.6 

4866.8 

8914.0 

9693.8 

2673.8 

5013.1 

9087.4 

32 

34 

9641.9 

2633.0 

4871.6 

8919.7 

9695.0 

2675.2 

.5018.0 

9093.2 

34 

36 

96-J3.7 

2634.4 

4876. 4 

8925.4 

9697.4 

2676.6 

5023.0 

9099.1 

36 

38 

9615.5 

2635.8 

4881.2 

8931.1 

9699.1 

2678.0 

5028.0 

9105  0 

38 

40 

9617.3 

2637.2 

4885.0 

8936.8 

9700.9 

2679.5 

5032.9 

9110.8 

40 

42 

96J9.1 

2638.6 

4890.9 

8942.6 

9702.7 

2680.9 

5037.9 

9116.7 

42 

44 

9650.9 

^640.0 

4895.7 

8948.3 

9704.5 

2682.3 

5042.9 

9122.6 

44 

46 

9652.7 

2641.4 

4900.5 

8954.0 

9706.2 

2683.7 

5047.9 

9128.5 

46 

48 

9654.5 

2642  9 

4905.3 

8959  8 

9708.0 

2685.1 

5052.9 

9134  4 

48 

50 

9656.3 

2644  3 

4910.2 

8965.5 

9709.8 

2686.5 

5057.9 

9140.3 

50 

52 

9658.1 

2645.7 

4915.0 

8971.3 

9711.6 

2687.9 

5062.9 

9146.2 

52 

51 

9659.9 

2647.1 

4919  9 

8977.0 

9713.3 

2689.3 

5067.9 

9152.1 

54 

56 

9661.7 

2648.5 

4924.7 

8982.8 

9715.1 

2690.7 

5072.9 

9158.1 

56 

58 

9663.5 

2649.9 

4929.6 

8988. 5 

9716.9 

2692.2 

5078.0 

9164.0 

58 

60 

9665.3 

2651  3 

4934.4 

8994.3 

9718.6 

2693.6 

5083.0 

9169.9 

60 

0 

llfi^ 

117° 

/ 
0 

L.  C. 

M. 

E. 

T. 

L.  C. 

M. 

E. 

T. 

9718.6 

2693.6 

5083.0 

9169.9 

9771.3 

2736.1 

5236.6 

9350.5 

•> 

9720.4 

2695.0 

5088  0 

9175.9 

9773.0 

2737.5 

5241.8 

9356.6 

2 

4 

9722.2 

2696.4 

5093.1 

9181.8 

9774.7 

2738.9 

5247.0 

9362.7 

4 

6 

9723.9 

2697.8 

5098.1 

9188.8 

9776.5 

2740.4 

.5252.2 

9368.9 

6 

8 

9725.7 

2699.2 

5103  2 

9193.7 

9778.2 

2741.8 

5257.4 

9375.0 

8 

10 

9727.4 

2700.6 

5108.2 

9199.7 

9779.9 

2743.2 

5262.6 

9381.1 

10 

12 

9729.2 

2702.1 

5113.3 

9205.6 

9781.7 

2744.6 

5267.9 

9387.3 

12 

14 

9731.0 

2703.5 

5118.4 

9211.6 

9783.4 

2746.0 

5273.1 

9393.4 

14 

16 

9732.7 

2704.9 

5123.4 

9217.6 

9785.2 

2747.5 

5278.4 

9399.5 

16 

18 

9734.5 

2706.3 

5128.5 

92:.'3.6 

9786.9 

2748.9 

5283.6 

9405.7 

18 

20 

9736.3 

2707.7 

5133.6 

9229  9 

9788.6 

2750.3 

5288.9 

9411.9 

20 

22 

9738.0 

2709.1 

5138.7 

9235  5 

9790.4 

2751.7 

5294.2 

9418.0 

22 

24 

9739.8 

2710.6 

5143.8 

9211.5 

9792.1 

2753.2 

5299.5 

9424.2 

24 

20 

9741.5 

2712.0 

5148.9 

9247.6 

9793.8 

2754.6 

5304.7 

9430.4 

26 

28 

9743. 3 

2713.4 

5154  0 

9253.6 

9795.6 

2756.0 

5310.0 

9436.6 

28 

30 

9745.0 

2714.8 

5159.1 

9259.6 

9797.3 

2757.4 

5315.3 

9442.8 

30 

32 

9746.8 

2716.2 

.5164.2 

9265  6 

9790.0 

27.58.9 

5320.6 

9449.0 

32 

34 

9748.5 

2717.6 

5169.4 

9271.6 

9800.7 

2760.3 

5325.9 

9455.2 

34 

36 

9750.3 

2719.1 

5174.5 

9277  7 

9802.5 

2761.7 

5331.2 

9461.4 

36 

38 

9752.0 

2720.5 

5179.7 

9283.7 

9804.2 

2763.1 

5336.5 

9467.6 

38 

40 

9753.8 

2721.9 

5184.8 

9289.8 

9805.9 

2764.6 

5341.8 

9473.8 

40 

42 

9755.6 

2723.3 

5190.0 

9295.8 

9807.7 

2766.0 

5847.2 

9480.0 

42 

44 

9757.3 

2724.7 

5195.1 

9301.9 

9809,4 

•767.4 

5352.5 

9486.3 

44 

46 

9759.0 

2726.2 

5200.3 

9307.9 

9811.1 

2768.8 

5357.9 

9492.5 

46 

48 

9760.8 

2727.6 

5205.4 

9314.0 

9812.8 

2770.3 

5363.2 

9498.7 

48 

50 

9762.5 

2729.0 

5210.6 

9320.1 

9814.5 

2771.7 

5368.5 

9505.0 

.50 

52 

9764.3 

2730.4 

5215.8 

9326.1 

9816.3 

2773.1 

5373.9 

9511.2 

52 

54 

9760.0 

2731.8 

5221.0 

9332.2 

9818.0 

2774.6 

5379.3 

9517.5 

54 

56 

9767.8 

2733.3 

.5226.2 

9338.3 

9819.7 

2776.0 

53S4.7 

9.523.8 

56 

58 

9769.5 

2734.7 

.5231.4 

9344.4 

9821.4 

2777.4 

5390.0 

9530.0 

58 

60 

9771.3 

2736.1 

.5236.6 

93.50.5 

9823.1 

2778.8 

5395.4 

9.536  3 

60 

298 


TABLE  X.— SINES  AND   COSINES. 


~0 

0°    1 

1°     1 

2°    1 

3°    1 

40 

60 

sine 

.ooooo; 

Cosin 
One. 

Sine 
.01745 

Cosin 
T99985 

Sine 
703490 

Cosin 

Sine 
.05234 

Cosin 
T99863 

Sine  Cosin 
.06976  .99756' 

.99939 

1 

.00029 

One. 

.01774 

.99984 

.03519 

.999:38 

.05263 

.99861 

.07005 

.99754 

59 

2 

.OOO08 

One. 

.01803 

.99984 

.03548 

.999:37 

.0.3292 

.99860 

.07034 

.997521 

58 

3 

.00087 

One. 

.01832 

.99983 

.03577 

.999)36 

.05:321 

.99858 

.070631 

.99750 

57 

4 

.00116 

One. 

.01862 

.99983 

.0:3606 

.999:35 

.05:350 

.99857 

.07092 

.99748' 

56 

5 

.00145 

One.  ; 

.01891 

.99982; 

;  .036:35 

.99934 

.05:379' 

.99855 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99982 

.0:3664 

.99933 

.05408 

.99854 

.07150 

.99744 

54 

7 

.00204 

One. 

.01949 

.999811 

!.  0:3693 

.999:32 

.05437 

.998521 

.07179 

.99742 

53 

8 

.00233 

One. 

.01978 

.999801 

.0:3723 

.999.31 

.05466 

.998.51 

.07208 

.99740 

52 

9 

.00262 

One. 

.02007 

.99980 

.0:3752 

.99930 

.05495 

.99849^ 

.072.37 

.99738 

51 

lO 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

j  .05524 

.99847: 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

'.03810 

.99927 

i  .05553 

.99846 

.07295 

.99734 

49 

12 

.00349  .99999 

.02094 

.99978 

.0:3839 

.99926 

1 .05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.999991 

.02123 

.99977 

.0:3868 

.99925 

! .05611 

.99842 

.07353 

.99729 

47 

14 

.00407 

.99999 

.021.52 

.99977 

.03897 

.99924 

.05640 

.99841 

.07:382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

; .03926 

.99923 

.05069 

.99839 

.07411 

.99725 

45 

16 

.00465 

.99999 

.02211 

.99976 

.0:3955 

.99922 

.05698 

.99838 

.07440 

.99723 

44 

17 

.00495 

.99999 

.02240 

.99975 

1.03984 

.99921 

.057'27 

.99836, 

.07469 

.99721 

43 

18 

.00524 

.99999: 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00.553 

.99998 

.02298 

.99974 

.04042 

.99918 

.057M5 

.998:33 

.07527 

.99716 

41 

20 

.90582 

.99998 

.02:327 

.99973 

.04071 

.999171 

! .05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02.356 

.99972 

.04100 

.99910 

' .05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.0.5873 

.99827 

.07614 

.99710 

38 

23 

.00669 

.99998 

.02414 

.99971 

1.04159 

.99913 

.05902 

.99826 

.('7643 

.99708 

37 

24 

.00698 

.99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824 

.07672 

.99705 

36 

25 

.00727 

.999971 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

.00756 

.99997. 

!  .02.501 

.99969 

.04246 

.99910 

.0.5989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.025.30 

.99968 

: .04275 

.99909 

.00018 

.99819 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

i  .04.304 

.99907 

.06047 

.99817 

.07788 

.99696 

32 

29 

.00844 

.99996 

.02589 

.99966 

.04.333 

.99906 

.00076 

.99815 

.07817 

.99694 

31 

30 

.00873 

.99996, 

.02618 

.99966 

.04362 

.99905 

'••  .06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

! .04391 

.99904 

.00134 

.99812 

.07875 

.99689 

29 

32 

.009-31 

.99996 

.02676 

.99964 

1 .04420 

.99902 

.06163 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.079:33 

.99685 

27 

34 

.00989 

.99995 

.027^4 

.99963 

.04478 

.999(W 

.06221 

.99806 

.07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

! .04507 

.99898 

i .06250 

.99804 

.07991 

.99080 

25 

36 

.01047 

.99995! 

.02792 

.99961 

*  .04.5:36 

.99897 

1 .06279 

.99803 

.08020 

.99678 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04.565 

1.99890 

.06308 

.99801 

.08049 

.99676 

23 

38 

.01105 

.999941 

.028.50 

.99959 

i  .04.594 

'.99894 

j  .06337 

.99799 

1 .08078 

.99673 

22 

39 

.01134 

.99994! 

.02879 

.99959 

.04023 

.99893 

.06:360 

.99797 

.08107 

.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

1.04653 

'.99892 

I .06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993! 

.029.38 

.99957 

''.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992! 

.02996 

.99955 

.04740 

.99888 

.00482 

.99790 

.08223 

.99661 

1  17 

44 

.01280 

.99992; 

\  .03025 

.999.54 

.04769 

.99880 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.999.53 

.04796 

.99885 

! .06540 

.997-86 

.08281 

.99657 

15 

46 

.01338 

.99991 

.03083 

.99952 

.04827 

.998a3 

.06.569 

.99784 

.08310 

.99654 

14 

47 

.01367 

.99991 

.03112 

.99953 

.04856 

.99882 

.00598 

.99782 

.08.339 

.99652 

13 

48 

.01396 

.99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.999901 

.03170 

.99950 

.04914 

'.99879 

'  .066.56 

.99778 

.08.397 

.99647 

11 

50 

.01454 

.99989 

.03199 

.99949 

.04943 

■ 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

. 99989 ' 

.03228 

.99948 

.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99989: 

.03257 

.99947 

.0.5001 

.99875 

.06743 

.99772 

.08484 

.996:39 

8 

53 

.01542 

.99988 

.03286 

.90946 

.0.5030 

.99873 

i  .06773 

.99770 

.08513 

.996.37 

7 

54 

01.571 

.99988! 

.03316 

.99945 

.050.59 

.99872 

.06802 

.99768 

.08542 

.996:35 

6 

55 

.01600 

.99987 

.0.3345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.996.32 

5 

56 

.01629 

.99987' 

.03374 

.99943 

1.0.5117 

.99S69 

.06860 

.99764 

.08600 

.996:30 

4 

57 

.01658 

.99986 

.0:3403 

.99942 

1 .0.5146 

.99867 

.06889 

.99762 

.08629 

.99627 

3 

58 

.01687 

.99986 

.0:34:32 

.99941 

.0.517-5 

.99S60 

1 .06918  .99760 

.086.58 

.99625 

2 

59 

.01716  .99985 

.03461 

.99940 

.0.5205 

.99864 

1  .0<i947  .997.58 

.08687 

.99622 

1 

60 

.01745  .99985 

.03490 

.99939 

i. 05234 

.99863 

.06976  .99756 

.08716 

.99619 

0 

/ 

Cosiu  1  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosiu  1  Sine 
86° 

Cosin 

^iae 

« 

8 

tjo 

8( 

50 

8' 

^ 

8^ 

>=> 

TABLE  X— SINES  AND  COSINES. 


299 


, 

5°    1 

6» 

70 

«•     1 

90 

f 

Sine 

Cosin 

Sine 

Co.sin 

Sine 

Cosin  ■ 

Sine  1 

Cosin 

!  Sine 

Cosin 

"o 

.08716 

799619 

.10453 

T99452 

T12187 

^9925,5 

.13917 

.99027 

.15643 

798769 

60 

1 

.08745 

.99617 

.10482 

.99449 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764 

59 

2 

.08774 

.99614 

.10511 

.994-16 

.12245 

.99248 

.13975 

.99019 

.15701 

.98760 

58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.987.55 

57 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.'99240 

.14033 

.99011 

! .1575? 

.98751 

56 

5 

.08800 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

! .15787 

.98746 

55 

6 

.08889 

.99604 

[.10626 

.99434 

.12360 

.99233 

.14090 

.99002 

.15818 

.98741 

.54 

7 

.08918 

.99602 

.106.-)5 

.99431 

.12389 

.992.30 

.14119 

.98998 

.1.5845 

.98737 

53 

8 

.08947 

.99.VJ9 

.  10(]S4 

.99428 

.12418 

.9i)226 

.14148 

.98994 

M5873 

.987:32 

52 

9 

.08976 

.99596 

.10713 

.99424 

.12417 

99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.1247T} 

! 99219 

.14205 

.98986 

' .15931 

.98723 

50 

11 

.09034' 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

1.15959 

.98718 

49 

12 

. 09063 : 

.99588 

. 10800 

.994151 

.12533 

.99211 ' 

.14263 

.98978 

'  .15988 

.98714 

48 

13 

.09092 

.99586 

.10829 

.99412 

.12562 

.99208' 

.14292 

.98973 

1.16017 

.98709 

47 

14 

09121 

.99583 

1 . 10858 

.99409' 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

46 

15 

09150 

.99580 

1.10887 

.99406 

.12620 

.99200 

.14340 

.98962 

.16074 

.98700 

45 

16 

.09179 

.99578 

:.  10916 

.99402 

.12649 

.99197 

.14.378 

.98961 

.16103 

.98695 

44 

17 

.09208 

.99575 

.10945 

.99399; 

.12678 

.99193 

.14407 

.98957 

.16132 

.98690 

43 

18  1 

.09237 

.99572 

.10973 

.99396' 

.12706 

.99189 

.14436 

.98953 

.16160 

.98686 

42 

19 

.09206 

.99570 

.11002 

.993931 

.12735 

.99186 

. 14464 

.98948 

.16189 

.98681 

41 

20 

.09295 

.99507 

.11031 

.993901 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

21 

.09324 

.99564 

L 11060 

.993861 

.12793 

.99178' 

.14522 

.98940 

.16246 

.98671 

39 

22 

.09353 

.99562 

.11089 

.99383 

.12822 

.99175 

.14551 

.98936 

.16275 

.98667  38  | 

23 

.09382 

.99559 

.11118 

.99380 

.128,51 

.99171; 

.14580 

.98931 1 

.16304 

.98662 

37 

24 

.09411 

.99556 

.11147 

.993771 

.12880 

.99167 

.14608 

.98927 

.16333 

.98657 

36 

25 

.09440 

.99.553 

.11176 

.993741 

.12908 

.99163 

.14637 

.98923 

.16361 

.986.52 

35 

26 

.09469 

.99.551 

.11205 

.99370' 

.12937 

.99160 

.14666 

.98919 

.16390 

.98648  34 

27 

.09498 

.99548 

.11231 

.99367! 

.12966 

.99156 

.14695 

.98914 

.16419 

.98643  33 

28 

.09527 

.99545 

AViijii 

.99304 

.12995 

.99152 

.14723 

.98910 

.16447 

.986:38  32 

29 

.09556 

.99.542 

.11291 

.99360 

.13024 

.99148 

.14752 

.98906 

.16476 

.986.33 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

30 

31 

.09614 

.99537 

.11349 

.99351 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

29 

32 

.09642 

.99534 

.11378 

.99351 

.13110 

.99137 

.14838 

.98893 

.16.562 

.98619 

28 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614 

27 

34 

.09700 

.99528 

.11436 

.99344, 

.13168 

.991291 

.14896 

.98884 

.16620 

.98609 

26 

35 

.09729 

.99526 

.11465 

.99341! 

.13197 

.991251 

.14925 

.98880 

1.16648 

.98604 

25 

36 

.09758 

.99523 

.11494 

.99337 

.13226 

.99122 i 

.14954 

.98876 

1.16677 

.98600 

24 

37 

.09787 

.99520 

.11.523 

.99334 

.132.54 

.99118! 

.14982 

.98871 

:. 16706 

.98595 

23 

38 

.09816 

.99.517 

.11552 

.99331, 

.13283 

.991141 

.15011 

.98867 

.167.34 

.98.590 

22 

39 

.09845 

.99514 

.11.580 

.99327 

.13.312 

.99110 

.1.5040 

.98863 

.16763 

.98585 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106: 

.15069 

.98858 

1.16792 

.98580 

20 

41 

;  09903 

.99.508 

.11638 

.99320 

.13370 

.99102' 

.1.5097 

.98854 

.16820 

.98575 

19 

42 

.09932 

.99.506 

1.11667 

.99317 

.13399 

.99098! 

.1.5126 

.98849 

!.  168-19 

.98570 

18 

43 

.09961 

.99503 

.11690 

.99314 

.13427 

.99094' 

.1.5155 

.98845 

.16878 

.98565 

17 

44 

.09990 

.99.500 

.11725 

.99310 

.134,58 

.99091 

.15184 

.98841 

.16906 

.98561 

16 

45 

.10019 

.99497 

.117.54 

•99307 

.i;i485 

.990871 

.15212 

.98836 

.169:35 

.98.5.56 

15 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.1.5241 

.98832 

.16964 

.98551 

14 

47 

. 10077 

.99491 

.11812 

.99300 

.13.543 

.99079 

.15270 

.98827 

1.16992 

.98546 

13 

48 

.10106 

.99488 

.11840 

.99297 

.13.572 

.99075! 

.15299 

.98823 

1 .17021 

.98.541 

12 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.1.5.327 

.98818 

I .17050 

.98.536 

11 

50 

.10164 

.99482 

.11898 

.99290 

.13629 

.99067 

.15356 

] .98814 

.17078 

.98531 

10 

61 

.10192 

! 99479 

.11927 

.99286 

.136.58 

.99063 

.1.5.385 

'.98809 

.17107 

.98526 

9 

52 

.10221 

.99470 

;.  11 9.50 

.99283 

.13687 

.99059 

.1.5414 

.98805 

.17136 

.98.521 

8 

53 

.10250 

.99473 

: .11985 

.99279 

.13716 

.990.55 

.1.5442 

.98800 

.17164 

.98516 

7 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

t.  990.51 

.1.5471 

.98796 

.17193 

.98.511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

1.99047 

.15.500 

.98791 

.17222 

.98.506 

5 

56 

.10337 

.99464 

.12071 

.99269 

.13802 

i. 99043 

.1.5.529 

.98787 

.172.50 

.98.501 

4 

57 

10366 

.99461 

.12100 

.99265 

.13831 

.99039 

.1.5.5.57 

.98782 

.17279 

.98496 

3 

58 

.10395 

.994.58 

.12129 

.99262 

.13860 

.99035 

.1.5.586 

.98778 

.17308 

.98491 

2 

59 

.10424 

.994.55 

.121.58 

.992.58 

.13889 

.99031; 

.1.5615 

.98773 

.17:3:36 

.98486 

1 

6< 

.10453 

.99452 

.12187 

.99255 

1.13917 

.99027 

.1.5643 

.98769 

:.  17365 

.98481 

^ 

1 

Cosin 

Sine 

Cosin 

Sine 

'  Cosin 

,  Sine" 

Cosin 

Sine 

1  Cosin 

Sine 

f 

84» 

83» 

82>» 

81° 

80» 

300 


Table  x.— sines  and  cosines. 


1, 

10° 

1    11° 

12° 

13° 

14°    !    1 

t 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine  ! 

Cosin 

~o 

.17365  .98481 

.19081 

.98163 

.20791  .97815 

.2-2495 

.974:37 

.24192 

.97030  60 

1 

.17393  .98476 

' .19109 

.98157 

.20820  .97809 

.2-252:3 

.974:30 

.24220 

.970-23  59 

2 

.17422  .98171 

.19138 

.98152 

.20848  .9780:3 

.2-2552 

.97424 

.24-249 

.97015  58 

3 

.17451  .98466 

.19167 

.98146 

.20877 

.97797 

.2-2.5«0 

.97417 

.24277 

.97008  57 

4 

.17479  .98461 

.19195 

.98140 

.20905 

.97791 

.2-2608 

.97411 

24305 

.97001  56 

5 

.17508  .98455 

.19224 

.98135 

.209:33 

.97784 

.2-2637 

.97404 

.24:333 

.96994  55 

6 

.17537  .98450 

.19252 

.98129 

.20962 

.97778 

.2-2665 

.07398 

.24:362 

.96987  54 

7 

.17565  .98445 

.19281 

.98124 

.20990 

.97772 

.2-2693 

.97:391 

.24:390 

.96980  53 

8 

.17594  .98440 

.19:309 

.98118 

.21019 

.97766 

.2-2722 

.97384 

.24418 

.96973  52 

9 

.1762:3  .984:35 

.193:38 

.98112 

.21047 

.97760 

.22750 

.97.378 

.24446 

.96966  51 

10 

.17651  .984:30 

.19:366 

.98107 

.21076 

.97754 

.22778 

.97:371 

.24474 

.96959  50 

11 

.17680 

.98425 

.19:395 

.98101 

.21104 

.97748 

.2-2807 

.97365 

.24503 

.96952  49 

12 

.17708 

.98420 

.19423 

.98096 

.21132 

.97742 

.228:35 

.97:358 

.245:31 

.96945  48 

13 

.17737 

.98414 

.19452 

.98C90 

.21161 

.97735 

.2-286:3 

.97351 

.24559 

.96937  47 

.  14 

.17766 

.98409 

.19481 

.98084 

.21189 

.97729 

.2-2892 

.97:345 

.24587 

.969:30  46 

15 

.17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.2-2920 

.97:338 

.24615 

.969-23  45 

16 

.17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.2-2948 

.973:31 

.24644 

.96916  44 

17 

.17852 

.98:394 

.19566 

.98067 

.21275 

.97711 

.2-2977 

.97325 

.24672 

.96909  43 

18 

.17880 

.98:389 

.19595 

.98061 

.21:303 

.97705 

.2.3005 

.97318 

.24700 

.96902  42 

19 

.17909 

.  98:38:3 

.1962:3 

.98056 

.21:3:31 

.97098 

.2:30:33 

.97:311 

.247-28 

.96894  41 

20 

.179:37 

.98:378 

.19652 

.98050 

.21:360 

.97692 

.23062 

.97304 

.247.56 

.96887  40 

21 

.17966 

.98373 

.196.80 

.98044 

.21:388 

.97686 

.23090 

.97298 

.247^ 

.96880  39 

22 

. 17995 

.98:368 

.19709 

.980:39 

.21417 

.97680 

.2:3118 

.97-291 

.24813 

.96873  38 

23 

.18023 

.98:362 

.197:37 

.980:3:3 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866  37 

24 

.18052 

.98:357 

.19766 

.98027 

.21474 

.97667 

.23175 

.97278 

.24869 

.968.58  36 

25 

.18081 

.98352 

.19794 

.98021 

.21502 

.97661 

.-23-203 

.97271 

.24897 

.96851  35 

26 

.18109 

.98;i47 

.19823 

.98016 

.21.5:30 

. 97655 

.23231 

.97264 

.24925 

.96.844  34 

27 

.18138 

.98^1 

.19851 

.98010 

.21559 

.97648 

.23260 

.97257 

i  .24954 

.968.37  .33 

28 

.18166 

.98:336 

.19880 

.98004 

.21.587 

.97642 

.23-288 

.97251 

;  .24982 

.968-29  32 

29 

. 18195 

.98331 

.19908 

.97998 

.21616 

.976:36 

.23:316 

.97244 

; .25U10 

.96822  31 

30 

.18224 

.98325 

.19937 

.97992 

.21644 

.97630 

.23345 

.97237 

'  .25038 

.96815  30 

31 

.18252 

.98320 

.19965 

.97987 

.21672 

.97623 

.23:373 

.97230 

.25066 

.96807  29 

32 

.18281 

.98315 

.19994 

.97981 

.21701 

.97017 

.2^01 

.972-23 

.25094 

.96800  28 

33 

.18309 

.98310 

.20022 

.97975 

.21729 

.97611 

.2i429 

.97217 

.25122 

.96793  27 

34 

.18338 

.98304 

.20051 

.97969 

.21758 

.97604 

.2^458 

.97210 

.25151 

.96786  26 

35 

.18367 

.98299 

.20079 

.97963 

.21786 

.97598 

.2^486 

.97203 

.25179 

.96778  25 

36 

.18:395 

.98294 

.20108 

.97958 

.21814 

.97592 

.-2-3.514 

.97196 

.25-207 

.96771  24 

37 

.18424 

.98288 

.201:30 

.97952 

.21843 

.97585 

.23.542 

.97189 

.25-235 

.96764  23 

38 

.18452 

.982*3 

.20165 

.97946 

.21871 

.97.579 

.-2:3571 

.97182 

.25-263 

.967.56  22 

39 

.18481 

.98277 

.20193 

.97940 

.21899 

.97573 

.23599 

.97176 

.25291 

.96749  21 

40 

.18509 

.98272 

.20222 

.979:34 

.21928 

.97560 

.23627 

.97169 

.25:320 

.96742  20 

41 

.18538 

.98267 

.202.50 

.97928 

.21956 

.97560 

.23656 

.97162 

.2.5348 

.96734  19 

42 

.18567 

.98261 

.20279 

.97922 

.2198.5 

.97553 

.23684 

.97155 

.25:376 

.96727  18 

43 

.18595 

.98250 

.20:307 

.97916 

.22013 

.97547 

.2:3712 

.97148 

.25404 

.96719  17 

44 

.18624 

.98250 

.20:3:36 

.97910 

.22041 

.97541 

.23740 

.97141 

.254:32 

.96712  16 

45 

.18652 

.98245 

.2a3t>4 

.97905 

.22070 

.97.534 

.23769 

.971:^ 

.25460 

.96705  15 

46 

.18681 

.98240 

.20:393 

.97899 

.22098 

.97.528 

.23797 

.97127 

.25488 

.96697  14 

47 

.18710 

.98234 

.20421 

.97893 

.22126 

.97521 

.2:3825 

.971-20 

.2.5516 

.96690  13 

48 

.18738 

.98229 

.204.50 

.97887 

.22155 

.97515 

.-2:3^53 

.97113 

.25545  .96682  12 

49 

.18767 

.9822:3 

.20478 

.97HS1 

.22183 

.97508 

oo^s.;-? 

.97106 

.25.573  .96675  11 

50 

.18795 

.98218 

.20507 

.97875 

.22212  .97502 

.'23910 

.97100 

.25601  .%667  10 

51 

.18824 

.98212 

.20535 

.97869 

.22240  .97496 

.239:38 

.97093 

.256-29  .96660  9 

52 

.18852 

.98207 

.20563 

.97863 

.22268  .97489 

.23966 

.97086 

.2.5657  .96653  8 

53 

.18881 

.98201 

.20592 

.978.57 

.22297 

.97483 

:  .23995 

.97079 

.25685  .96645  7 

54 

.18910 

.98196 

.20620 

.97851 

.22:325 

.97476 

.240-23 

.97072 

.2.5713  .966:38  6 

55 

.18938 

.98190 

.20649 

.97.845 

.22:3.5:3 

.97470 

.-24051 

.97065 

;  .25741  .966:30  5 

56 

.18967 

.98185 

.20677 

.978:39 

.22:382 

.97463 

.-24079 

.97058 

'  .25769  .966-23  4 

57 

.18995 

.98179 

.20706 

.9783:3 

.22410 

.97457 

.24108 

.97051 

: .25798  .96615  3 

58 

.  190*24 

.98174 

.207:34 

.97827 

.224:38 

.97450 

.241:36 

.97044 

,  .258-26  .96608  2 

59 

.19052 

.98168 

.2076:3 

.97821 

.22467 

.97444 

.241W 

.97(^37 

'  .'25854  .96600  1 

60 

.19081 

'.9816:3 

.20791 

.97815 

.2-2495 

.97437. 

1  .24192 

.97030 

.25882  .96593  0 

< 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

i  Cosin  i  Sine 

'  Cosin  Sine 

/ 

79° 

1    78° 

1    77° 

76° 

75° 

TABLE  X.-SINES   AND  COSINES. 


301 


15°   1 

1    16°    1 

1    17°    1 

1    18° 

19° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

"o 

.25882 

796593 

.27564 

.96126 

.29237 

795630 

.30902 

.9.5106 

.32.557 

794T552 

60 

1 

.25910 

.96585 

.27592 

.96118 

.29265 

.95622 

.30929 

.95C:)7 

.32584 

.94542 

.59 

2 

.25938 

.96578 

.27620 

.96110 

.29293 

.95613 

.30957 

.9.5088 

.32612 

.94533 

58 

3 

.25966 

.96570 

.27648 

.96102 

.29321 

.95605 

.30985 

.95079 

.32639 

.94523 

57 

4 

.25994 

.96562 

.27676 

.96094 

, .29348 

.95596 

.31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.27704 

.96086 

1 .29376 

.95588 

.31040 

.95061 

.32694 

.94504 

55 

6 

.26050 

.96547; 

.27731 

.96078 

.29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.26079 

.96540' 

.27759 

■96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.26107 

.9()532 

.27787 

.96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.96524 

.27815 

.9()().54 

.29487 

.95554 

.31151 

.95024 

.32804 

.9446(i 

51 

10 

.26163 

.96517 

.27843 

. 96046 

.29515 

.95545} 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191 

.96509' 

.27871 

.960371 

1 .29543 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.9.5528 

.31233 

.94997 

.32887 

.?>um 

48 

13 

.26247 

.96494 

.27927 

.96021' 

.29599 

.95519 

.31261 

.94988 

.32914 

.94428 

47 

14 

.26275 

.96486 

.27955 

.96013 

.29626 

.95511 

.31289 

.94979 

.32942 

.94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316 

.94970 

.32969 

.94409 

45 

16 

.26331 

.96471 

.28011 

.95997 

1 .29682 

.95493 

.31344 

.94961 

.32997 

.94399 

44 

17 

.26359 

.96463 

.28039 

.95989 

'  .29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

18 

.2638? 

. 96456 

.28067 

.959811 

.29737 

.9:5476 

.31399 

.94943 

.33051 

.94:380 

42 

19 

.26415 

.96448 

.28095 

.95972 

.29765 

.95467 

.31427 

.94933 

1 ,33079 

.94370 

41 

20 

.2644:3 

.96440 

.28123 

.95964 

'  .29793 

.95459 

.31454 

.94924 

.33106 

• 

.94:361 

40 

21 

.26471 

.96433 

.28150 

.959.56 

.29821 

.95450' 

:  .31482 

.94915 

.33134 

.94351 

39 

22 

.26500 

.96425 

.28178 

.95948 

.29849 

.9.5441 

1 .31510 

.94906 

.33161 

.94342 

38 

23 

.26528 

.96417 

.28206 

.95940 

.29876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

24 

.26556 

.96410 

.28234 

.95931 

.29904 

.954241 

' .31565 

.94888 

.33216 

.94:322 

36 

25 

.26584 

.96402 

.28262 

.95923 

: .29932 

.95415; 

; .31593 

.94878 

.33:^4 

.94313 

35 

26 

.26612 

.96394 

.28290 

.95915 

.29960 

.95407 

.31620 

.94869 

.33271 

.94303 

34 

27 

.26640 

.96386 

.28318 

.95907 

: .29987 

.95398 

.31048 

'.94860 

.33298 

.94293 

33 

28 

.26668 

.96379 

.28346 

.95898 

.30015 

.95389 

i  .31675 

.94851 

.33326 

.94281 

32 

29 

.26696 

.96371 

.28374 

.95890 

.30043 

.95380 

j  .31703 

.94842 

.333.53 

.94274 

31 

30 

.26724 

.96363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

■  .33:381 

.94204 

30 

31 

.26752 

.96355 

.28429 

.95874 

.30098 

.95363 

.31758 

.94823 

'  .33408 

.94254 

29 

32 

.26780 

.96347 

.28457 

.95865 

! .30126 

.95354' 

.31786 

.94814 

.33436 

.94245 

28 

33 

.26808 

.96340, 

.28485 

.95857 

i  .30154 

.95345 

.31813 

.94805 

.33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225. 

26 

35 

.26864 

. 96324 i 

.28541 

.95841 

.30209 

.95328 

.31868 

.94786 

.33518 

.94215 

25 

36 

.26892 

.96316 

.28569 

.95832 

.30237 

.95319 

.31896 

.94777 

.33545 

.94206 

24 

37 

.26920 

.96308 

.28597 

.95824 

.30265 

.95310 

.31923 

.94768 

.3:3573 

.94196 

23 

38 

.26948 

.96301 

.28625 

.95816 

.30292 

.95301 

.31951 

.94758 

;  .;33600 

.94186 

22 

3!) 

.26976 

.96293 

.28652 

.9.5807 

.30320 

.95293 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27004 

.96285 

.28680 

.95799 

.30348 

.95284 

1 

.32000 

.94740 

.33655 

.94167 

20 

41 

.27032 

.9627?' 

.28708 

.95791 

.30376 

.95275 

! .32034 

.947'30 

.33082 

.941.57 

19 

42 

.27060 

.96269: 

.28736 

.95782 

.30403 

.95266! 

1 .32061 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261 

.28764 

.95774 

' .30431 

.95257 

; .32089 

.94712 

1 .a3737 

.941:37 

17 

4-1 

.27116 

.96253 

.28792 

.95766 

! .30459 

.95248 

i  .32110 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28820 

.95757 

.30480 

. 95240 i 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

28847 

.95749 

.3051> 

.95231 

.32171 

.94681 

! .33819 

.94108 

14 

47 

.27200 

.96230; 

."28875 

.95740 

.30542 

.952221 

.32199 

.94674 

' .33846 

.94098 

13 

48 

.27228 

.96222 

.28903 

.95732 

; .30570 

.952131 

.32227 

.94665 

i .33874 

.94088 

12 

49 

.27256 

.96214 

.28931 

.95724 

1 .30597 

.95204 

.32254 

.946.56 

3:3901 

.94078 

11 

50 

.27284 

.96206 

.28959 

.95715 

.30625 

.95195 

.32282 

.94646 

.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

.30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698] 

.30680 

.95177 

.32337 

.94627 

.33983 

.94049 

8 

53 

.27368 

.96182 

.29042 

.95690 

.30708 

.95168 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396 

.96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94(109 

..^038 

.94029 

6 

55 

.27424 

.96166 

.29098 

.95673 

.307'63 

.95150 

.32419 

.94.599 

.34065 

.94019 

5 

56 

.27452 

.96158 

.29126 

.95664 

.30791 

.95142 

.32447 

.94590 

.340931 

.94009 

4 

57 

.27480 

.96150 

.291,54 

.95656 

.30819 

.95133 

.32474 

.9.3.5801 

.;34120 

.93999 

3 

58 

.27508 

.96142 

.2!)! 82  .95647 

.;30846 

.951^ 

.32.502 

.94571 

.:34147 

.9:3989 

2 

59 

.27536 

.961.^ 

.29209  .9.5639, 

.30874 

.95115 

.32529 

.94:561 

.34175, 

.9:39791 

1 

60 

.27564 

.96126 

.29237 

.9.5630 

.:30902i 

.95106 

.32557 

.94.552 

.34202! 

.939691 

0 

/ 

Cosin 

Sine  1 

C3osin ' 

Sine 

Cosin  Sine 

Cosin 

Sine 

Cosin  1  Sine 

74»    1 

73° 

72°   i 

71°   1 

to° 

303 


TABLE  X.-SINES  AND  COSINES. 


r 

20° 

2] 

■"        1 

22 

! 

23 

°   il 

24°   1 

Sine 

Cosin 

Sine  Cosin ! 

Sine  Cosin ! 

Sine  Cosin 

Sine  Cosin 

/ 

"o 

^34202 

.93969 

.3.58:37  . 9:3.358  i 

.;37461 

.92718 

.3907:3  .92050; 

.4067'4  .913.55'  60 

1 

.34229 

.939.59 1 

.35864  .9:31*48 ; 

.37488 

.92707 

.39100' 

.92039! 

A0700   .91:343 

59 

2 

.34257 

.93949: 

.35891  .93:3:371 

.37515. 

.92697; 

.39127' 

.92028: 

.407'27  .91:331 

58 

3 

.34284 

.93939! 

.35918 

.9:3:327 

.37542! 

.92686; 

.391.53! 

.92016; 

.40753  .91319 

57 

4 

.34311 

.93929; 

.35945 

.93:316 

.37569 

.92675 

.39180' 

.92005' 

.40780  .91307 

56 

& 

.34339 

.939191 

.35973 

.93:306 

.37595 

.926641 

.39207! 

.919941 

.40806  .91295 

55 

6 

.34366 

.939091 

.36000 

.9:3295 

.37622; 

.92653 

.392.34 

.91982; 

.40833  .91283 

54 

7 

.34393 

.93899 

.:36027 

.9:3285 

..37649 

.92642 

.39260 

.919711 

.40860  .91272 

53 

8 

'.34421 

.93889; 

.:36054 

.9.3274 

.37676' 

.92631 

.39287 

.919591 

.40886  .91260 

52 

9 

.34448 

.93879 

.,36081 

.9:3264 

.37703 

.92620 

.39:314 

.91948' 

.40913 

.91248 

51 

10 

.34475 

.93869 

.36108 

.9:3253 

.37730 

.92609 

.39341 

.91936; 

.409.39 

.91236 

50 

11 

..34.503 

.93859 

.36135 

.9.3243 

.01 lOl 

.92598 

.39367 

.91925 

.40966' 

.91224 

49 

12 

.34.5.30 

.93849 

.36162 

.9:32.32 

.37784 

.92587 

..39.394 

.91914 

.40992 

.91212 

48 

13 

.34.557 

.93839; 

.36190 

.9:3222 

.37811 

.92576 

•  .:3942I 

.91902; 

.41019 

.91200; 

47 

14 

.34.584 

.938291 

.36217 

.0:3211 

.  3  ( 8:38 

.92565 

.;39448 

.91891 1 

.41045 

.91188 

46 

15 

.34612 

.93819 

.36244 

.9:3201; 

. 37865 

.92554 

.39474 

.918791 

.41072 

.91176 

46 

IG 

..34639 

.93800! 

.36271 

*.93190 

.37892 

.92543 

.39501 

.91868' 

.41098 

.91164 

44 

17 

..34666 

.9.3799; 

.36298 

.93180 

.37919 

.92532; 

.39528 

.91856! 

.41125 

.91152 

43 

18 

.34694 

.93789 

.36325 

.93169 

.37946 

.92521; 

1  ..S9.555 

.91845; 

.41151 

.91140  42 

19 

.34721 

.93779 

'  .36:3.52 

.93159 

.37973 

.92510: 

'  .39.581 

.9ia33' 

.41178 

91128 

41 

iiiO 

.34748 

.93769, 

j  .36.379 

.93148: 

.37999 

.92499 

1  .39608 

.91822^ 

.41204 

.91116 

40 

21 

.34775 

.9.375©! 

1 .36406 

.93137; 

..38026 

.92488 

'  .396.35 

.91810 

.412.31 

.91104 

39 

22 

..34803 

.93748 

..364:34 

.931271 

.38053 

.92-i77 

.39661 

.91799 

' .41257 

.91092  .38 

23 

..348.30 

.937.38 

.3(>461 

.931101 

.38080 

.92466, 

.39688 

.91787 

; .41284 

.91080  37 

24 

.34857 

.93728 

.36488 

. 93106 1 

.38107 

.92455' 

.3971.5, 

.91775 

1 .41310 

.91068  :36 

25 

.34884 

.93718 

1 .36515 

.93095, 

.aSi34 

.92444 

.39741 

.917(>4 

.41.337 

.91056  35 

26 

.34912 

.9.3708 

.36542 

.9:3081 

.:38101 

.924:32 

.39768 

.91752 

:  .41.363 

.91044  34 

27 

.349.39 

.9.3698 

..36569 

.93074 

.38188 

.9242. 

t  .39795 

.91741 

.41390 

.910:32  33 

28 

.34966 

.9.3688 

.36.596 

.9.3063 

.38215 

.92410 

.:39822 

.91729 

.41416 

.91020  .32 

29 

.34993 

.93677 

.36623 

.930.521 

.:-:8241 

.92:399 

..3984!^ 

.91718 

.41443 

.91008  31 

30 

.35021 

. 93667 1 

.36650 

.93042 

.38268 

.92.388 

,.39875 

.91706 

: .41469 

.90996  30 

1 

31 

.35048 

.9.3657 

'  ..36677 

.9:3031 

..38295 

.92377 

!  .;399<}2 

.91694 

.41496 

.90984  29 

32 

.35075 

.93647 

..36704 

.9:3020 

.:38322 

.92:366 

.39928 

.91688 

.41522 

.90972  28 

33 

.35102 

.93637 

.36731 

.9:3010 

.38:349 

.92:355 

.39955 

.91671 ' 

.41549 

.90960  27 

34 

..35130 

.93626 

.:367.58 

.92999 

.38376 

.92:343 

.39982 

.9I6(;0 

.41575 

.90948 

26 

35 

.36157 

.93616 

..36785 

.92988' 

.38403 

.92:332 

.40008 

.91648 

.41602 

.909:36 

25 

36 

.35184 

.93606 

;  ..36812 

.92978 

.384:30 

.92:321 

.400:35 

.91636 

.41628 

.90924 

24 

37 

.3.5211 

9.3596 

:  ..368:39 

.92967 

.:3a456 

.92.310 

.40062 

.91625 

.416.55 

.90911 

23 

38 

.. 3.52:39 

.93.585 

..36867 

.929.56 

.38483 

.92299 

' .40088 

.91613 

.41681 

! .90899 

22 

39 

.3.5266  .93.575 

..36891 

.92945 

..38510 

.92287 

!  .40115 

.91601 

.41707 

.90887 

21 

40 

.35293 

.9.3565 

..36921 

.92935 

.38537 

.92276 

!  .40141 

.91590 

.41734 

.90875 

1 

20 

41 

.a5.320 

.9.3.555 

..36948 

.92924 

'  .3856-1 

'.92265 

' .40168 

.91.578 

'  .417'60 

.90863  19 

42 

..3.5347 

.93544 

,  .:36975 

.92913 

.38.591 

.922.54 

.40195 

.91566 

.47787 

.90851  18 

43 

.  .35375 

.93534 

.:37002  .92902 

.  .38617 

.92243 

.40221 

L 91555 

.41813 

.908:39  17 

44 

..3.5402 

.93.5.'^! 

.37029 

.92892 

..38f)}4 

.92231 

:  .40.248 

.91.543 

.41840 

.90826  16 

45 

..35429 

.9.3514 

.370.56 

.92881 

'  ..38671 

.92220 

.40275 

;.  91.5:31 

.41866 

.90814  15 

46 

.354.56 

.93503 

.37083 

.92870 

..38698 

.9^209 

.#198 

.40:301 

!.  91.519 

.41892 

;. 90802 

14 

47 

.3^K4 

.9:^^493 

.37110 

.92a59 

..•i8725 

.40328 

.91.508 

.41919 

.90790 

13 

48 

. 3551 1 

.93483 

.3;  1:37  .9:^849 

, 00  t 04 

.92180 

.40:355 

.91496 

.41945 

.90778 

12 

49 

..3.5.538 

.93472 

..37164 

.928:38 

.38778 

.92175 

.40381 

1.91484 

.41972 

.90766 

11 

50 

.35565 

.93462 

.37191 

.92827 

.38805  .92164 

.40408  .91472 

.41998 

_.  90753 

10 

51 

.35592 

.93452 

..37218 

.92816 

.388.32  .921.52 

i  .40434  .91461 

.42024 

'  .90741 

9 

52 

..3.5619 

.93441 

..37245 

.92805! 

..388.59  .92141 

•  .40461  .91449 

.42051 

.90729 

I  8 

53 

.3.5647 

.9:3431 

.:37272 

.92794 

..38886  .921.30; 

'  .40488  .91437 

.42077 

!. 90717 

:  7 

54 

..35574 

.93420 

.37299 

.92784; 

; .38912  .92119! 

1  .40.514  .91425 

.42101 

'.907'n4'  6 

55 

..3.5701 

.9.3410 

.37326 

.92773 

..389:39  .92107; 

i  .40.541  .91414 

.421.30 

.90692  5 

56 

.35728 

.9.3400 

..37:353 

.92762 

..38966  .92096' 

!  .40.567  .91402 

.421.56 

.90680 

1  4 

57 

..3.57.55 

.93.389 

.:37.380 

.92751 i 

.38993  .92085 

.40.594  .91390 

.42183 

.90068 

.  3 

58 

.35782 

.9.3:379 

.37407 

.92740 

.:39O20  .9207-31 

.40621  .91.378 

42209 

.906,55 

0 

59 

..35810 

.9.3:368 

..37434 

.92729' 

..39046  .92062 

.40647  .91366 

.42235 

.90643 

1 

60 

.358-37 

.93358 

.37461 

.92718 

..39073  .920.50 

.40674  .913.55 

.42262 

.90631 

li 

Cosiu 

Sine  1 

Cosin 

Sine 

Cosin  Sine  | 

Cosin  Sine 

Cosiu 

1  Sine 

/ 

69°    ! 

68' 

!    67° 

66° 

65° 

.'ABLE   X.-SINES  AND  COSINES. 


303 


/ 

25° 

26°    1 

27° 

28°      : 

29° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin ' 

Sine 

Cosin 

"o 

.422(52 

.90631 

.4:3837 

789879 

.4.5399 

.89101 

1  .4(5947 

788295 

.48481 

T87462 

60 

1 

.42288 

.90618 

.43863 

.89807 

.45425 

.89087 

.46973 

.88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

58 

3 

.42341 

.90.594 

;  .4:3916 

.89841 

.45477 

.89061 

.47024 

.882541 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45.503 

.89048 

.47050 

.88240; 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43968 

.898161 

.45.529 

.890:35 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42120 

.90557 

.4:39i)4 

.898031 

.455.54 

.89021 

.47101 

.88213 

.48634 

. 87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.4,5.580 

.89008 

.47127 

.88199 

.48659 

.87.363 

53 

8 

.42473 

.90.532 

.44046 

.89777 

.45606 

.88995 

.471.53 

.88185 

.48684 

.87.349 

52 

9 

.42199 

.90520 

.44072 

.89764 

.45(5:32 

.88981 

.47178 

.88172 

.48710 

.873,35 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87.321 

50 

11 

.425.52 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87.306 

49 

12 

.42578 

.90183 

.44ir.i 

.89726 

.45710 

.83942 

.472.55 

.88130 

.48786 

.87292 

4i^ 

1-3 

.42604 

.90470 

.44177 

.80713 i 

.457:36 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45;'62 

.88915 

.47306 

.88103 

.488:37 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.904:33 

.44255 

.89674 

.45813 

.88888 

.473.58 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.896631 

.4.58:39 

.88875 

1  .47.383 

.88062 

1 .48913 

.87221 

43 

18 

.42736 

.90408 

.44:307 

.89649' 

.4.5865 

.88862 

' .47409 

.88048 

1 .48938 

.87207 

42 

19 

.42762 

.90:396 

:  .44:3:53 

.896:36! 

.45891 

.88848 

.474:34 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623: 

.45917 

.88835 

.47460 

.88020 

' .48989 

.87178 

40 

21 

.42815 

.90:371 

.44.385 

.89610; 

.45942 

.88822 

1.47486 

.88006 

i  .49014 

.87164 

39 

22 

.42841 

. 90358 

.44411 

.89597 

.4.5968 

.88808 

.47.511 

.87993 

.49040 

.871.50 

38 

23 

.42867 

.90346 

.444:37 

.89.584' 

.4.5994 

.88795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.903:34 

.44464 

.89.571- 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90:321 

.44490 

.89.558; 

.46046 

.88768 

1 .47588 

.87951 

.49116 

.87107 

35 

2H 

.42946 

.90309 

.44516 

.89.5451 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44.542 

.89.532 

.46097 

.88741 

.476:39 

.87923 

.49166 

.87079 

33 

2S 

.42999 

.90284 

.44508 

.89519 

' .46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.91271 

.44.594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.870.50 

31 

30 

.43051 

.902.59 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87030 

30 

31 

.4.3077 

.90246 

.44046 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.902:33 

.44672 

.89407 

.46226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.46252 

.88661 

.47793 

.87840 

.49.318 

.8(5993 

27 

34 

.43156 

.90208 

.44724 

.89441 

.46278 

.886471 

.47818 

.87826 

.49.344 

.86978 

26 

35 

.43182 

.901961 

.44750 

.89428 

.46.3(34 

.88(5.34' 

.47844 

.87'812 

.49369 

.86964 

25 

3(5 

.4.3209 

.90ia3 

.44776 

.89415 

.46:3.30 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

'  .4(5:355 

.88607 

.47895 

.877-84 

.49419 

.86935 

23  1 

38 

.43261 

.90158 

.44828 

.89:389 

.46.381 

.88593 

.47920 

.87770 

.49445 

.8(5921 

22  ! 

39 

.43287 

.90146 

.448.54 

.89:376 

.46407 

.88.580 

.47946 

.877.56 

.49470 

.8(5906 

21  1 

40 

.43313 

.901:33, 

.44880 

.89:363 

.46433 

.88566 

.47971 

.87743 

.49495 

1 

.86392 

20  1 

41 

.43.340 

.90120 

.44906 

.89350 

.464.58 

.88553 

.47997 

.877-29 

1 .49521 

.86878 

19  ' 

42 

.4:!366 

.90108 

.449:32 

.89.337 

.4648-1 

.885.39 

.48022 

.87'715 

.49546 

.86863 

18  ' 

43 

.4.i392 

.90095 

.449.58 

.89:324 

.4(5510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.4?>418 

.90082 

.44984 

.89311 

.46.5:36 

.88512 

.4807'3 

.87-G87 

.49596 

.80831 

16 

45 

4;M45 

.90070 

.4.5010 

.89298 

.46.561 

.88499 

.48099 

.87-673 

.49622 

.86820 

15 

46 

.4:3471 

.900.57 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.4:3497 

.900151 

.4.5062 

.89272 

.40613 

.88472 

.48150 

.87'645 

.49672 

.80791 

13 

48 

.4:3523 

.900:32 

.4.5088 

.89259 

.466:39 

.88458 

.48175 

.87631 

.49697 

.8677V 

12 

49 

.4.3.549 

.90019 

.45114 

.89245 

.4<i664 

.884-15 

!  .48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007) 

.45140 

.892:32 

.46690 

.88431 

1  .48226 

.87603 

! .49748 

.86748 

10 ; 

51 

.43602 

.89994' 

.45166 

.89219 

.46716 

.88417 

.48252 

.87.589 

.49773 

.867,33 

9 

52 

.4:3628 

.89981 

.45192 

.89200, 

.46742 

.88404 

1  .48277 

.  87575 

: .49798 

.86719 

8 

53 

.4:3654 

.89968 

.4.5218 

.89193; 

.46767 

.8839,) 

.48:303 

.87-.561 

' .49F24 

.86704 

7 

54 

.4:3680 

.899.56 

.4.5243 

.89180! 

.46793 

.8a377 

.48.328 

.87-516 

.49849 

.86690 

6 

55 

.4.3706 

.8994:3 

.4.5269 

.89167 

.46819 

.88363 

'  .48:3.54 

.87-5:32 

.49874 

.86675 

5 

56 

.43733 

.899:30 

.45295 

.891.53 

;  .4(5814 

.8,8:349 

.48:379 

.87.518 

i  .49899 

.86661 

4 

57 

.4:i7.59 

.89918 

.45321 

.89140! 

.4(5870 

.8K-3:36 

1  .4^405 

.87.504 

!  .49924 

.86646 

3 

58 

.4:5785 

.89905 

.45.347 

.89127 

.4(5896 

.88322 

.484:30 

.87490 

.499.50 

.866:32 

2  : 

59 

.4:3811 

.89892 

.4.5:373 

.89114 

.4(5921 

.88:308 

1  .484.56 

.87-476 

.49975 

.86617 

1 

60 

.4:38:371.89879 

.4.5:399 

.89101 

.4(5947 

.88295 

1 .48481 

.87402 

..50000 

.86603 

_0 

/ 

Cosin  i  Sine 

Cosin 

Sine 

Cosin 

Sine 

1  Cosin 

j 

Sine 

Cosin 

Sine 

/ 

S4° 

^3° 

62° 

'   61" 

'        60° 

304 


TABLE  X. -SINES  AND  COSINES. 


1 

80°    1 

31°    11 

32° 

33°    j 

34*    1 

f 

Sine 

Cosin 

Sine 

Cosin 

Sine  1 

Cosin 

Sine  1 

Cosin 

Sine  1 

Cosin 

"o 

Tsoooo 

.86603 

^51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919! 

.82904' 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017 

.84789 

.54488 

.83851 

.55943 

.82887 

59 

2 

.50050 

.86573 

.S1554 

.85687 

.530411 

.84774 

.54513 

.83835 

.55968 

.82871 1 

58 

3 

.50076 

.86559 

..51579 

.85672 

.53066 

.84759 

.54537! 

.83819 

.55992 

.828551 

57 

4 

.50101 

.86544 

.51604 

.85657 

.5.0091 

.84743 

.54561 

.83804 

.56016 

.82839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

..56040 

.82822 

55 

6 

.50151 

.86515 

.51653 

.85627 

.53140 

.84712 

.54610 

.83772 

.56064 

.82806 

54 

7 

.50176 

.86501 

.51678 

.85612 

.531641 

.84697 

.54635; 

.83756 

.56088 

.82790 

53 

8 

..50201 

.86486 

..51703 

.85597 

.53189' 

.84681 

.54659' 

.83740 

.56112 

.82773 

52 

9 

.50-:?27 

.80171 

.51728 

.8.5,582 

.53214' 

.84666 

.54683 

.83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.837u8 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635' 

.54732 

.8.3692 

.56181 

.82724 

49 

12 

.50302 

.80427 

.51803 

.8,5533 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708 

48 

13 

..50327 

.86413 

.51828 

. 85521 

.53312 

.84604 

.54781 

.83660 

.56232 

.82692 

47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675 

46 

15 

.50377 

.86.384 

.51877 

.85491 

.53361 

.84573 

..54829 

.83629 

.56280 

.82659  45 

16 

.50403 

.86369 

.51902 

.85476 

.5.3.386 

.84557 

.548.54 

.83613 

.56305 

.82643  44 

17 

.50428 

.863.54 

.51927 

.8.5461 

.5.3411 

.84542 

.54878 

.8.3597 

.56329 

.82626  43 

18 

.50453 

.86340 

.519.52 

.85446 

.5.34.35 

.84526 

.54902 

.8.3581 

.56353 

.82610  42 

19 

..50478 

.86325 

.51977 

.8.5431 

.53460 

.84511 

..';4927 

.83.565 

.56377 

.82.593  41 

20 

.50503 

.86310 

.52002 

.85416 

,53484 

.84495 

.54951 

.83549 

.56401 

.82577  40 

21 

..•50528 

.86295 

.52026 

.85401 

.53509 

. 84480 ' 

.54975 

.83533 

.56425 

.82.561'  39 

22 

.50.553 

.86281 

1  ..52051 

.85.385 

..53534 

.84464 

.54999 

.8.3517 

.56440 

.82544  38 

23 

.50578 

.86266 

! .52076 

.85370 

.5.3558 

.84448 

.55024 

.83501 

.56473 

.82528  37 

24 

.50603 

.86251 

.52101 

.853.55 

..5.3583 

.84433 

.55048 

.83485 

.56497 

.82511  36 

25 

..50628 

.862.37 

.52126 

.8.5.340 

.5.3607 

.84417 

.55072 

.8.3469 

.56521 

.82495  .35 

26 

.506.54 

.86222 

..521.51 

.8,5325 

.53632 

.84402 

.55097 

.8:M53 

.56545 

.82478  34 

27 

.50679 

.86207 

..52175 

.85310 

.53056 

.84:386 

.55121 

.834.37 

.56,569 

.82462  33 

28 

..50704 

.86192 

..52200 

.8.5294 

.53681 

.84:370 

.5,5145 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

..52225 

.85279 

.53705 

.84:355 

.55169 

.8:3405; 

.56617 

.82429 

31 

30 

.50754 

.86163 

.522.50 

.85264 

.53730 

.84:339 

.55194 

.83389^ 

.56641 

.82413 

30 

31 

.50779 

.86148 

..52275 

.85249 

.5-37.54 

.84.324 

.55218 

.8.3373 

.56665 

.82396 

29 

32 

..50804 

.86133 

.:yi-im 

.852.34 

.53779 

.84:308 

.55242 

.8.3356 

.56689 

.82380 

28 

33 

.50829 

.86119 

.,52324 

.852181 

.53804 

.84292 

..55266 

.83340 

.56713 

.82.363 

27 

34 

..50854 

.86104' 

.,52349 

.8,5203! 

.53828 

.84277 

.5.5291 

.8.33^ 

.567.36 

.82347 

26 

35 

.50879 

.86089 

.52:374 

.8.5188 

.5.3853 

.84261 

.55315 

.8.3308 

.56760 

.82330 

25 

36 

..50904 

.86074 

.52399 

.851731 

.53877 

.84245 

.5,53.39 

.83292 

.56784 

.82.314  24 

37 

.50929 

.80059 

..52423 

.8,51.57 

.53902 

.842:30 

.5.5.363 

.83276 

.56808 

.82297  23 

38 

.509.54 

.86045 

..52448 

.85142 

.5.3926 

.84214 

.5.5388 

.8.3260 

.56832 

.82281 

23 

39 

..50979 

.86030, 

..52473 

.85127 

.5.3951 

.84198 

.5.5412 

.83^4 

.56856 

.82264 

21 

40 

.51004 

8G015 ' 

.52498 

.85112 

.5.3975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000'; 

.52,522 

.85096 

.54000 

.84167 

.55460 

.83212 

' .56904 

.822.31 

19 

42 

.510.54 

.8.5985 

..52547 

.85081 

..54024 

.84151 

.5.5484 

.83195 

1 .56928 

.82214 

i  18 

43 

.51079 

.8.5970 

.52572 

.8,5066 

.54049 

.841.35 

.55509 

.83179 

' .56952 

.82198 

,17 

44 

.51104 

.859.56, 

.52.597 

.85051 

.54073 

.84120 

.5b533 

.8.3163 

.56976 

.82181 

'  16 

45 

..51129 

.85941 : 

.52621 

.8,50.35 

..54097 

.84104 

. 55557 

.83147 

.57000 

.82165 

15 

46 

.51154 

.85926 

..52646 

.85020 

.54122 

.84088; 

.55,581 

.8.3131 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.57047 

.82132 

;  13 

48 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

.57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.57095 

.82098 

11 

50 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.5.5678 

.83066 

.57119 

.82082 

10 

51 

.51279 

.85851 

.,52770 

.84943 

..54244 

.84009 

.55702 

.83050 

.57143 

.82065 

1  9 

52 

.51304 

.8.5836 

.,52194 

.84928 

.54269 

1-8.3994 

..55726 

.8:30.34 

.57167 

.82048 

8 

53 

.51329 

.8.5821 

.52819 

.S4913 

..54293 

1.839781 

.5.57,50 

.8.3017 

1 .57191 

.82032 

7 

54 

.51354 

.8.5806 

.52844 

.84897 

.W317 

.83962 

..55775 

.8:3001 

1  .57215 

.82015 

i  6 

55 

.51379 

.85792 

.52869 

.84882 

.54:342 

.8,3946 

.55799 

.82985 

.57238 

.81999 

5 

56 

.51404 

.8.5777 

..52S93 

.84866 

.54366 

.8.39.30 

.5.5823 

.82969 

.57262 

.81982 

4 

57 

.51429 

.8.5762 

.52918 

.84851 

.54391 

1.8:3915 

..5.5847 

.829.53 

i  .57286 

.81965 

3 

58 

.51454 

.8.5747 

.52943 

.84836 

..54415 

.83899 

.55871 

.829.36 

! .57310 

.81949  2 

59 

.51479 

.8.57.32 

.52967 

.84820 

.54440 

.8:3883; 

.55895 

.82920 

!  .57a34 

.81932  1 

60 

.51.504 

.8,5717 

.52992 

.84805 

.54464 

.83867 

( .55919 

.82904 

i. 57358 

.81915  0 

/ 

Cosin 

Sine 

Cosin 

1  Sine 

Cosin 

i  Sine" 

Cosin 

Sine 

Cosin 

Sine 

1 

59° 

58° 

57° 

56° 

55° 

«-i 

TABLE   X. -SINES   AND   COSINES. 


305 


/ 

35°    1 

36°   1 

37°    1 

38°    1 

39° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

1 
Cosin 

Sine 

Cosin 

Sine 

Cosin 

"o 

.57358 

.81915 

.58779 

.80902! 

760182 

:79864 ' 

761.566 

77'8801 

.62932 

.77715 

60 

1 

.57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

59 

2 

.57405 

.81882' 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678 

58 

3 

.57429 

.818651 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660 

57 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.77641 

50 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

55 

6 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694 

.6.3068 

.77605 

54 

7 

.57524 

.81798 

.58943 

.80782' 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548 

.81782 

.58967 

.80765' 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57.572 

.81765 [ 

.58990 

.80748! 

.60.390 

.79706! 

.61772 

.78640' 

.63135 

.77550 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.796881 

.61795 

.78622 

.63158 

.77531 

50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.796711 

.61818 

.78604 

.63180 

.77513 

40 

12 

.57643 

.81714 

..59061 

.80096; 

.60400 

.79653! 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679! 

.60483 

.79635 

.61864 

.7'8568 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80662 

.00506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532: 

.63271 

.77439 

45 

16 

.57738 

.81647 

.59154 

.80627' 

.60553 

.79583 

.619.32 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

..59178 

.80610 

.60576 

.79505 

.61955 

.78496 

.6.3316 

.77402 

43 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338 

.77384 

42 

19 

.57810 

.81597 

.59225 

.80570 

.60622 

.79530 

.62001 

.78460 

.63.361 

.77366 

41 

20 

.57833 

.81580 

.59248 

.80558, 

.60645 

.79512 

'  .62024 

.78442 

.63383 

.77347 

40 

21 

.57857 

.815631 

.50272 

.80541' 

.60668 

.79494' 

.62046 

.78424' 

.63406 

.77329 

39 

22 

.57881 

.81546! 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310 

38 

23 

.57904 

.81530 

.59.318 

.80507 

.60714 

.79459; 

.62092 

.78387 

.83451 

.77292 

37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.62115 

.78369 

.63473 

.77273 

36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424! 

.62138 

.78351 

.6*496 

.77255 

35 

26 

.57976 

. 81479 1 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.814621 

.59412 

.804.38 

.00807 

.79388 

.62183 

.78315 

.6.3540 

.77218 

33 

28 

.58023 

.814451 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297! 

.63563 

.77199 

32 

29 

.58047 

.81428! 

.59459 

.80403 

60853 

.79353 

i  .62229 

.78279! 

.63585 

.77181 

31 

30 

.58070 

.81413 

.59482 

.80386 

.60876 

.79335 

j  .62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80.368 

.60899 

! 79318 

.62274 

.78243 

.63630 

.77144 

29 

32 

.58118 

.81378: 

.59529 

.80.3.51 

.60922 

.79300! 

; .62297 

.78225; 

.63653 

.77125 

28 

33 

.58141 

.813611 

.59552 

.80334 

.60945 

. 79282 1 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81.344! 

.59576 

.80316 

.60968 

.79264' 

.62342 

.78188: 

.63698 

.77088 

26 

35 

.58189 

.813271 

.59599 

.80299' 

.60991 

.79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

..58236 

.81293! 

.59646 

.80264 

.010.38 

.79211! 

! .62411 

.78134 

.63765 

.77033 

23 

38 

.58260 

.81276! 

.59669 

.80247 

.61061 

.79193! 

1 .62433 

.78116- 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.01084 

.79176 

.62456 

.78098! 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.01107 

.79158, 

.62479 

.78079: 

.63832 

.76977 

20 

41 

.58330 

.812251 

.59739 

.80195! 

.61130 

.79140' 

.62502 

.78061' 

.63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122; 

i  .62524 

.78043 

.63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 1 

.61176 

.791051 

.62547 

.78025 

.63899 

.76921 

17 

44 

.58401 

.81174 

.59809 

.801431 

.61199 

.79C87 

.62.570 

.78007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59832 

.80125; 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

.58449 

.81140 

.598.56 

.80108; 

.61245 

.79051 

[ .62615 

.77970 

.63966 

.76866 

14 

47 

.58472 

.811231 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

-.80073! 

.61291 

.79016 

! .62660 

.77934' 

.64011 

.76828 

12 

49 

.58519 

.81089 

.59926 

.800.56 

.61314 

.78998: 

: .62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.800381 

.61337 

.78980; 

.62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021! 

.61360 

.78962 

' .62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.81038 

.59995 

.80003 

.61383 

.78944, 

.62751 

.77861, 

.64100 

.76754 

8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843; 

.64123 

.76735 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.70717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

5 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

. 77788 

.64190 

.76679 

4 

57 

.58708 

.80953 

.60112 

■.79916 

.61497 

.78855' 

.62864 

.77769 

.64212 

.76661 

3 

58 

.58731 

.809.36' 

.60135 

.79899 

.61520 

.78837 

.62887 

.77751) 

.64234 

.76642 

2 

59 

.587.55 

.80919 

.60158 

.79881 

.61543 

.78819 

.62909 

.77733 

.64256 

.76623 

1 

60 

..58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62^32 

.77715 

.64279 

.76604 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

64° 

53° 

52° 

61» 

50° 

306 

TABLE  X.- 

-SINES  AND 

COSINES. 

/ 

40° 

41" 

42° 

43° 

44° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

"o 

.64279 

.76604 

.65606 

.75471 

766913 

.74314 

768200 

.73135 

.69466 

.71934 

60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.04346 

.76548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

.G4368 

.70530 

.65094 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.04390 

.70511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833 

55 

6 

.04412 

.76492; 

.65738 

.75356 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

,657.59 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772 

52 

9 

.64479 

.70436! 

.6.5803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.04.524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

li 

.64546 

.78380 

.65Sj9 

.7.5241 

.67172 

.74080 

.68455 

.72897 

.69717 

.71691 

48 

13 

.64538 

.7(3361 

.6.5891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.76-i42[ 

.6.3913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

.64635 

.76304 

.65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71010 

44 

17 

.64657 

.76286 

.6.5978 

.75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590 

43 

IS 

.64679 

.76287 

.66000 

.75126 

.67301 

.73963 

.68582 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76248 

.66022 

.75107, 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229^ 

.66044 

.75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

.75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.66088 

.75050; 

.67387 

.73885 

.68666 

.72697 

.69925 

.71488 

38 

23 

.64790 

.76173 

.68109 

.75030' 

.67409 

.73865 

.68688 

.72677 

.69946 

.71468 

37 

24 

.64812 

.76154 

.66131 

.75011  1 

.67430 

.73846 

.68709 

.72657 

.69966 

.71447 

36 

25 

.64834! 

.76135 

.661.53 

.749921 

.67452 

.73826 

.68730 

.72637 

.69987 

.71427 

35 

26 

.64856 

.76116 

.66175 

.74973! 

.67-473 

.73806 

.68751 

.72617 

.70008 

.71407 

34 

27 

.64878; 

.76097 

.66197 

.74953' 

.67495 

.73787 

.68772 

.72597 

..70029 

.71386 

33 

28 

.649011 

.76078 

.66218 

.74934! 

.67516 

.73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.64923 

.76059 

.66240 

.74915 

.67538 

.73747 

.68814 

.72557 

.70070 

.71345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66284 

.74876' 

.67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989; 

.76003 

.68W6 

.74857 

.67602 

.73688 

.68878 

.72497] 

70132 

.71284 

28 

33 

.65011' 

.7.5984 

.66327 

.74838 

.67623 

.73669 

.68899 

.72477! 

.70153 

.71264 

27 

34  1 

.65033 

.75965 

.66349 

.74818 

.67645 

.73649 

.68920 

.72457! 

.70174 

.71243 

26 

35  I 

.65055 

.75946 

.66371 

.74799 

.67666 

.73629 

.68941 

.72437 

.70195 

.71223 

35 

36 

.650771 

.75927 

.66393 

.74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

37 

.  6.5100 : 

.75908 

.60414 

.74760 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

38  ! 

.65122; 

.7.5889 

.66436 

.74741 

.67730 

73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.651441 

.75870 

.684.58 

.74722 

.67752 

.73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851, 

.66480 

.74703 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

.758;i2 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

. 65210 i 

.75813 

.66523 

.74664 

.67816 

.73491 

.69088 

7OOQ7 

.70339 

.71080! 

18 

43 

.652321 

.75:94 

. 68545 

.74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059, 

17 

44 

.65254 

.75775 

.66566 

.74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

.75756 

.66.588 

.74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019! 

15 

46 

.65298 

.75738 

.66610 

.74586 

.67901 

.73413 

.69172 

.72216 

.704221 

.70998! 

14 

47 

.65320 

.75719 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.704431 

.70978' 

13 

48 

.65342 

.7.5700 

.066.53 

.74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.709.57; 

12 

49 

.65364 

.7.5680 

.66675 

.74528 

.67965 

.73353 

.69235 

.72156 

.704841 

.70937 

11 

50 

.65386 

.75661 

1 

.66697 

.74.509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916, 

10 

51 

.65408 

.75642 

.66718 

.7.4489 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896' 

9 

52 

.65430' 

.75623 

.66740 

.74470 

.68029 

.73294 

.69298 

.72095 

.70546  .70875 

8 

53 

.65452 

.7.5604 

.66762 

.74451 

.68051 

.73274 

.69319 

.72075 

.70567.708.55:  7 

54 

.65474 

.75.585 

.66783 

.74431 

.68072 

.73254 

.69340 

.72055; 

.70587  .70834'  6 

55 

.6.5496 

.75.566 

.66805  .74412 

.68093 

.73234 

.69361 

.720351 

.70608  .70813  5 

56 

.6.5518 

.75.547 

.66827  .74392 

.68115 

.73215 

.69382 

.72015 

.70628. 70793  4 

57 

.6.5.540 

.7.5.528 

.66848  .74373 

.68136 

.73195 

.69403 

.71995. 

.70649 

.70772  3 

58 

. 65562 

.7.5509 

.66870  .743.53 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752  2 

59 

.65584 

.75490 

.66891  .743:34 

.68179 

.73155 

i  .69445 

.71954 

.70690 

.70731 

1 

60 

.65606 

.75471 

.66913  .74314 

.68200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 

/ 

Cosin 

Sine 

Cosin  Sine 

1  Cosin 

1 

Sine 

Cosin 

Sine 

Cosin  1  Sine 

/ 

49°    1 

48° 

1    470 

46°    1 

i    45° 

XI.     NATURAL  SECANTS 

AND  COSECANTS. 

307 

/ 

0 

SECANTS. 

- 

0° 

1» 

2° 

3° 

4° 

6° 

6° 

1.00000 

1.00015 

1.00061 

1.00137 

1 .00244 

1.00382 

1 .00551 

60 

] 

00000 

00016 

00062 

00139 

00246 

00385 

00554 

59 

t> 

00000 

00016 

00063 

00140 

00248 

00387 

00557 

58 

3 

00000 

01K)17 

00064 

00142 

00250 

00390 

00560 

57 

4 

00000 

00017 

00065 

00143 

00252 

00392 

00563 

56 

5 

t)0000 

00018 

00066 

00145 

00254 

00395 

00566 

55 

f) 

00000 

00018 

00067 

00147 

00257 

00397 

00569 

54 

t 

00000 

00019  , 

00068 

00148 

00259 

00400 

00573 

53 

8 

00000 

00020 

00069 

00150 

0026 1 

00403 

00576 

52 

9 

00000 

00020 

00070 

00151 

00263 

00405 

00579 

51 

10 

00000 

00021 

00072 

00153 

00265 

00408 

00582 

50 

11 

1.00001 

1.00021 

1.00073 

1.00155 

1.00267 

1.00411 

1.00585 

49 

12 

00001 

00022 

00074 

00156 

00269 

00413 

00588 

48 

13 

00001 

00023 

00075 

00158 

00271 

00416 

00592 

47 

14 

00001 

00023 

00076 

00159 

00274 

00419 

00595 

46 

15 

00001 

00024 

00077 

00161 

00276 

00421 

00598 

45 

16 

00001 

00021 

00078 

00163 

00278 

00424 

00601 

44 

17 

00001 

00025 

00079 

00164 

00280 

00427 

00604 

43 

18 

00001 

00026 

00081 

00166 

00282 

00429 

00608 

42 

19 

00002 

00026 

00082 

00168 

00284 

00432 

00611 

41 

20 

00002 

00027 

00083 

00169 

00287 

00435 

00014 

40 

21 

1.00002 

1.00028 

1.00084 

1.00171 

1.00289 

1.00438 

1.00617 

39 

22 

00002 

00028 

00085 

00173 

00291 

00440 

00621 

38 

23 

00002 

00029 

00087 

00175 

00293 

00443 

00624 

37 

24 

00002 

00030 

00088 

00176 

00296 

00446 

00627 

36 

25 

00003 

00031 

00089 

00178 

00298 

00449 

00630 

35 

26 

00003 

00031 

00090 

00180 

00300 

00451 

00034 

34 

27 

00003 

00032 

00091 

00182 

00302 

00454 

00037 

33 

28 

00003 

00033 

00093 

00183 

00305 

00457 

00640 

32 

29 

00004 

00034 

00094 

00185 

00307 

00460 

00644 

31 

30 

00004 

00034 

00095 

00187 

00309 

00463 

00647 

30 

31 

1.00004 

1.00035 

1.00097 

1.00189 

1.00312 

1.00465 

1.00650 

29 

SZ 

00004 

00036 

00098 

001  !)0 

00314 

00468 

00654 

28 

33 

00005 

00037 

00099 

00192 

00316 

00471 

00657 

27 

34 

00005 

00037 

00100 

00194 

00318 

00474 

00660 

26 

35 

00005 

00038 

00102 

00196 

00321 

00477 

00064 

25 

36 

00005 

00039 

00103 

00198 

00323 

00480 

00667 

24 

37 

00U06 

00040 

00104 

00200 

00326 

00482 

00671 

23 

38 

00006 

00041 

00106 

00201 

00328 

00485 

00674 

22 

39 

00006 

00041 

00107 

00203 

00330 

00488 

00677 

21 

40 

00007 

00042 

00108 

00205 

00333 

00491 

00681 

20 

41 

1.00007 

1.00043 

1.00110 

1.00207 

1.00335 

1.00494 

1.00684 

19 

42 

00007 

00044 

00111 

00209 

00337 

00497 

00688 

18 

43 

00008 

00045 

00113 

00211 

00340 

00500 

00691 

17 

44 

00008 

00046 

00114 

00213 

00342 

00503 

00695 

16 

45 

00009 

00047 

00115 

00215 

00345 

00506 

00698 

1.5 

46 

00009 

00048 

00117 

00216 

00347 

00509 

00701 

14 

47 

00009 

00048 

00118 

00218 

00350 

00512 

00705 

13 

48 

00010 

00049 

00120 

00220 

00352 

00515 

00708 

12 

49 

00010 

00050 

00121 

00222 

00354 

00518 

00712 

11 

50 

00011 

00051 

00122 

00224 

00357 

00521 

00715 

10 

51 

1.00011 

1.00052 

1.00124 

1.00226 

1.00359 

1.00524 

1.00719 

9 

52 

00011 

00053 

00125 

00228 

00362 

00527 

00722 

8 

53 

00012 

00054 

00127 

00230 

00364 

00530 

00726 

54 

00012 

00055 

00128 

00232 

00367 

00533 

00730 

6 

55 

00013 

00056 

00130 

00234 

00369 

00536 

00733 

5 

56 

00013 

00057 

00131 

00236 

00372 

00539 

00737 

.4 

57 

00014 

00058 

001.33 

00238 

00374 

00542 

00740 

1 

58 

00014 

00059 

00134 

00240 

00377 

00545 

00744 

59 

00015 

00060 

00136 

00242 

00379 

00548 

00747 

T 

60 

00015 

00061 

00137 

00244 

00382 

00551 

00751 

0 

f 

89° 

88° 

87° 

86° 

85° 

84° 

83° 

/ 

COSECANTS. 

308       XI. -NATURAL   SECANTS   AND  COSECANTS. 


/ 

SECANTS. 

/ 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

0 

1.00751 

1.00983 

1.01247 

1.01543 

1.01872 

1.02234 

1.02630 

60 

1 

00755 

00987 

01251 

01.548 

01877 

02240 

02687 

59 

2 

00758 

00991 

012.56 

01553 

01883 

02247 

02644 

58 

3 

00762 

00995 

01261 

01.5.58 

01889 

02253 

02051 

57 

4 

00765 

00999 

01205 

01564 

01895 

02259 

02658 

56 

5 

00769 

01004 

01270 

01569 

01901 

02266 

02665 

55 

6 

00773 

01008 

01275 

01.574 

01906 

02272 

02672 

54 

7 

00776 

01012 

01279 

01.579 

01912 

02279 

02679 

53 

8 

00780 

01016 

01281 

01585 

01918 

02285 

02686 

52  ' 

9 

00784 

01020 

01289 

01590 

01924 

02291 

02693 

51 

10 

00787 

01024 

01294 

01595 

01030 

02298 

02700 

50 

11 

1.00791 

1.01029 

1.01298 

1.01601 

1.01936 

1.02304 

1.02707 

49 

12 

00";  95 

01033 

01303 

01606 

01941 

02311 

02714 

48 

13 

00799 

01037 

01308 

01011 

01947 

02317 

02721 

47 

14 

00802 

01011 

01313 

01616 

01953 

02323 

02728 

46 

15 

00806 

01046 

01318 

01622 

019.59 

02330 

02735 

45 

16 

00810 

01050 

01322 

01627 

01965 

02336 

02742 

44 

17 

00813 

01054 

01327 

01633 

01971 

02313 

02749 

43 

18 

00817 

01059 

01332 

01038 

01977 

02349 

02756 

42 

19 

00821 

01063 

01337 

01643 

01983 

02356 

02763 

41 

'JO 

00825 

01067 

01342 

01649 

01989 

02362 

02770 

40 

21 

1.00828 

1.01071 

1.01316 

1.01654 

1.01995 

1.02369 

1.02777 

39 

22 

00832 

01076 

01351 

016.59 

02001 

02375 

02784 

38 

23 

00836 

01080 

013.56 

01665 

02007 

023S2 

02791 

37 

24 

00840 

01084 

01361 

01670 

02013 

02388 

02799 

36 

25 

00844 

01089 

01366 

01676 

02019 

02395 

02806 

35 

26 

00848 

01093 

01371 

01681 

02025 

02402 

02813 

34 

27 

00851 

01097 

01376 

01687 

02031 

02408 

02820 

33 

28 

00855 

01102 

01381 

01692 

02037 

02415 

02827 

32 

29 

00859 

01106 

01386 

01698 

02043 

02421 

02834 

31 

30 

00863 

01111 

01391 

01703 

02049 

02428 

02842 

30 

31 

1.00867 

1.01115 

1.01395 

1.01709 

1.02055 

1.02435 

1.02849 

29 

32 

00871 

01119 

01-100 

01714 

02061 

02441 

02856 

28 

33 

00875 

01124 

01405 

01720 

02067 

02418 

028C3 

27 

34 

00878 

01128 

01410 

01725 

02073 

024.54 

02870 

26 

35 

00882 

01133 

01415 

01731 

02079 

02461 

02878 

25 

36 

00S86 

01137 

01420 

01736 

(.2085 

02468 

028S5 

24 

37 

0i890 

01142 

01425 

01742 

02091 

02474 

02892 

23 

38 

00894 

01146 

01430 

01747 

02097 

02481 

02899 

22 

39 

00898 

01151 

01135 

01753 

02103 

024S8 

02907 

21 

40 

00902 

01155 

01410 

017.58 

02110 

02494 

02914 

20 

41 

1.00906 

1  01160 

1.01445 

1.01764 

1.02116 

1.02501 

1.02921 

19 

42 

00910 

01164 

OI4.0O 

01769 

02122 

02508 

02928 

18 

43 

00914 

01169 

01455 

01775 

02128 

02515 

02936 

17 

44 

00918 

01173 

01461 

01T81 

02134 

02521 

02913 

16 

45 

00922 

01178 

01466 

01786 

02140 

02528 

029.50 

15 

46 

00926 

01182 

01471 

01792 

02146 

02535 

02958 

14 

47 

00930 

01187 

01476 

01798 

02153 

02542 

02965 

13 

48 

009:^4 

01191 

01481 

01803 

02159 

02548 

02972 

12 

49 

0093S 

01196 

014-6 

01809 

02165 

02555 

02980 

11 

50 

00942 

01200 

01491 

01815 

02171 

02562 

02987 

10 

51 

1.00946 

1.01205 

1.01496 

1.01820 

1.02178 

1.02.569 

1.02994 

9 

52 

00950 

01209 

01501 

01826 

02184 

02576 

03002 

8 

53 

00954 

0121 

01506 

01832 

02190 

02582 

03009 

7 

54 

00958 

01219 

01512 

01837 

02196 

02589 

03017 

6 

55 

00962 

012:3 

01517 

01843 

02203 

02596 

03024 

5 

56 

00966 

01228 

01522 

01849 

02209 

02603 

03032 

4 

57 

00970 

01233 

01527 

018.14 

02215 

02610 

03039 

3 

58 

00975 

01237 

01532 

01860 

02221 

02617 

03046 

0 

59 

00979 

01242 

01.537 

01866 

02228 

02624 

0:^054 

I 

60 

00983 

01247 

01543 

01872 

02234 

02630 

03061 

0 

/ 

82° 

81° 

80° 

79° 

78° 

77° 

70° 

/ 

COSECANTS. 

XI.— NATURAL  SECANTS  AND  COSECANTS.       309 


/ 
0 

SECANTS. 

/ 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

1.03061 

1.03528 

1.04030 

1.04.569 

1.05146 

1.0.5762 

1.06418 

60 

1 

03069 

03536 

04039 

04578 

051,56 

05773 

06429 

59 

o 

03076 

03544 

04047 

04588 

051G6 

05783 

06440 

58 

3 

030S4 

03552 

04056 

04.597 

05176 

05794 

06452 

57 

4 

03091 

03560 

04065 

04606 

05186 

05805 

06463 

56 

5 

03099 

03568 

04073 

04616 

05196 

0.5815 

06474 

55 

6 

03106 

03576 

040S2 

04025 

05206 

05826 

06486 

54 

i 

03114 

03584 

04091 

04635 

0.5216 

05836 

06497 

13 

8 

03121 

03592 

04100 

04641 

05226 

05847 

08.508 

52 

9 

03129 

03601 

01108 

04653 

05236 

05858 

06520 

51 

10 

03137 

03609 

04117 

01663 

05246 

05869 

06531 

50 

11 

1.03144 

1.03617 

1.04126 

1.01672 

1.052.56 

1.0.5879 

1.06542 

49 

1'^ 

03152 

03625 

04135 

04682 

05266 

05890 

06554 

48 

Vi 

031.59 

03633 

04144 

04691 

05276 

05901 

06565 

47 

11 

03167 

03642 

04152 

04700 

05286 

0.5911 

06577 

46 

15 

03175 

03650 

04161 

04710 

05297 

05922 

065S8 

45 

10 

03182 

03658 

04170 

04719 

05307 

05933 

06600 

44 

IT 

03190 

03666 

04179 

04729 

05317 

05944 

06611 

43 

18 

03198 

03674 

04188 

04738 

05.327 

05955 

06622 

42 

1!) 

03-J05 

03683 

04197 

04748 

0.5337 

0.5965 

06634 

41 

20 

03il3 

03691 

04206 

04757 

0.5347 

05976 

06645 

40 

21 

1.03221 

1.03699 

1.04214 

1.01767 

1.0.53.^7 

1.05987 

1.066.57 

39 

2i 

03228 

03708 

.04223 

04776 

05367 

0.5998 

06668 

38 

23 

03236 

03716 

04232 

04786 

05378 

06009 

06680 

37 

21 

03244 

03724 

04241 

04795 

05388 

06020 

06691 

36 

25 

03251 

03732 

042.50 

04805 

05398 

06030 

06703 

35 

2G 

032.59 

03741 

04259 

04815 

05408 

06041 

06715 

34 

27 

03:67 

03749 

04268 

01824 

0.5418 

06052 

06726 

33 

28 

03-^75 

03:5s 

04277 

04834 

05429 

06063 

00738 

32 

29 

03282 

03766 

042S6 

04843 

05439 

06074 

06749 

31 

30 

03290 

03774 

04295 

04853 

05449 

06085 

06761 

30 

31 

1 .03298 

1.0.3783 

1.04.304 

1.04863 

1.0.5400 

1.06096 

1.06773 

29 

32 

03306 

03791 

04313 

04872 

05470 

06107 

06784 

28 

33 

03313 

03799 

04322 

04882 

05180 

06118 

06796 

27 

34 

03321 

03808 

04331 

04891 

05490 

06129 

06807' 

26 

35 

03329 

03816 

01340 

04901 

05501 

06140 

06819 

25 

3(5 

03337 

03825 

04349 

04911 

05511 

06151 

06831 

24 

37 

03345 

03833 

04358 

04920 

0.5521 

06162 

06843 

23 

38 

03353 

0.3842 

04367 

04930 

05532 

06173 

06854 

22 

39 

03360 

03850 

04376 

04940 

05542 

06184 

06866 

21 

40 

03368 

03858 

04385 

049.50 

05552 

06195 

06878 

20 

41 

1.0.3376 

1.0.3867 

1.04394 

1.04959 

1.0.5.503 

1.06206 

1.06889 

19 

42 

03384 

03875 

04403 

04969 

05573 

06217 

06901 

18 

43 

03392 

03884 

04413 

04979 

05584 

06228 

06913 

17 

44 

03400 

03892 

04422 

04989 

0.5594 

06239 

06925 

16 

45 

03408 

03901 

01431 

04998 

05604 

06250 

069.36 

15 

4G 

03416 

03909 

04140 

05008 

0.5615 

06261 

06948 

14 

47 

03424 

0.3918 

04449 

05018 

05625 

06272 

06960 

13 

48 

03432 

03927 

04458 

0502S 

05636 

06283 

06972 

12 

49 

0.3439 

03935 

04468 

05038 

0.5646 

06295 

06984 

11 

50 

03447 

03944 

04477 

05047 

05657 

06306 

06995 

10 

51 

1.0.34.55 

1  03952 

1.04486 

1 .050.57 

1.05667 

1.06317 

1.07007 

9 

52 

03463 

03961 

04495 

05067 

05678 

06328 

07019 

8 

53 

03471 

03969 

04504 

05077 

05fi88 

06339 

07031 

54 

03179 

03978 

04514 

05087 

05699 

06350 

07043 

6 

55 

03487 

03987 

01523 

05097 

05709 

06362 

07055 

5 

5(5 

03495 

03995 

04.532 

05107 

05720 

06373 

07067 

4 

57 

03503 

04004 

04541 

05116 

05730 

063S4 

07079 

3 

58 

03512 

04013 

04551 

05126 

05741 

06395 

07091 

2 

59 

03520 

04021 

04560 

05136 

0.5751 

06407 

07103 

1 

60 

03528 

04030 

04569 

05146 

05762 

06418 

07115 

0 

/ 

75° 

74° 

7S° 

72° 

71° 

70° 

69° 

f 

COSECANTS. 

5iO       XI.— XATUHAL  SECANTS  AND  COSECANTS. 


1 

/ 

SECANTS. 

"1 

/ 

21° 

22° 

23° 

24° 

25° 

20° 

27° 

0 

1.07115 

1.07853 

1.08636 

1.09464 

1.10338 

1.11260 

1.12233 

eo 

1 

07126 

07866 

08649 

09478 

10353 

11276 

12249 

59 

>) 

07138 

07879 

08663 

09492 

10368 

11292 

12266 

58 

3 

07150 

07892 

08070 

09506 

1038:! 

11308 

12283 

4 

07162 

07904 

08690 

09520 

10398 

11323 

12299 

56 

5 

07174 

07917 

08703 

09535 

10413 

11339 

12316 

55 

6 

07 186 

07930 

08^17 

09.549 

10428 

11355 

12333 

54 

f 

( 

07199 

07943 

08730 

09563 

10443 

11371 

12349 

53 

8 

07211 

07955 

08744 

09577 

10458 

11.387 

12366 

52 

9 

07223 

07968 

08757 

09592 

10473 

11403 

12383 

51 

10 

07235 

07981 

08771 

09606 

10488 

11419 

12400 

50 

H 

1.07247 

1  07994 

1.08784 

1.09620 

1.10503 

1.11435 

1.12416 

49 

12 

07259 

08006 

08798 

09035 

10518 

11451 

12433 

48 

13 

07271 

08019 

08811 

09649 

10.533 

11467 

12450 

47 

14 

07283 

08032 

08825 

09663 

10549 

11483 

13467 

46 

15 

07295 

08045 

08839 

09678 

10564 

11499 

124S4 

45 

16 

07307 

08058 

08852 

09692 

10579 

11515 

12501 

44 

17 

07320 

08071 

08866 

09707 

10594 

11.531 

12518 

43 

18 

07332 

08084 

08880 

09721 

10609 

11547 

12534 

42 

19 

07344 

08097 

08893 

09735 

10025 

11563 

12551 

41 

20 

07356 

08109 

08907 

09750 

10640 

11579 

12568 

40 

21 

1.07368 

1.08122 

1.08921 

1.00764 

1.106.55 

1.11595 

1.12585 

39 

22 

07380 

08135 

08934 

09779 

10070 

11611 

12602 

38 

23 

07393 

08148 

08948 

09793 

10686 

11627 

12619 

37 

24 

07405 

08101 

08962 

09808 

10701 

11643 

12636 

36 

25 

07417 

08174 

08975 

09822 

10716 

11659 

12653 

35 

26 

07429 

08187 

08989 

09837 

10731 

11675 

12670 

34 

27 

07442 

08200 

09003 

09851 

10747 

11691 

12687 

33 

28 

07454 

08213 

09017 

09866 

10762 

11708 

12704 

32 

29 

07466 

08226 

09030 

098S0 

10777 

11724 

12721 

31 

30 

07479 

08239 

09044 

09895 

10793 

117-10 

12738 

30 

31 

1.07491 

1.08252 

1.090.58 

1 .09909 

1.10808 

1.11756 

1.12755 

29 

32 

07503 

08205 

09072 

09924 

10824 

11772 

12772 

28 

33 

07516 

08278 

09086 

09939 

10839 

11789 

12789 

27 

34 

07528 

08291 

09099 

09953 

10854 

11805 

12807 

26 

35 

07540 

08305 

09113 

09908 

10870 

11821 

12824 

25 

36 

07553 

08318 

09127 

09982 

10885 

11838 

12841 

24 

37 

07505 

08331 

09141 

09997 

10901 

118.54 

12858 

23 

.38 

07578 

08344 

09155 

10012 

10916 

11870 

12875 

22 

39 

07590 

08357 

09169 

10026 

10932 

11886 

12892 

21 

40 

07602 

083^0 

09183 

10041 

10947 

11903 

12910 

20 

41 

1.07615 

1.08383 

1.09197 

1.10055 

1.10963 

1.11919 

1.12927 

19 

42 

07627 

0S397 

09211 

10071 

10978 

11936 

12944 

18 

43 

07640 

08410 

09224 

10085 

10994 

11952 

12961 

17 

44 

07652 

08423 

09238 

10100 

11009 

11968 

12979 

16 

45 

07665 

08436 

09252 

10115 

11025 

11985 

12996 

15 

46 

07677 

08449 

09266 

10130 

11041 

12001 

13013 

14 

47 

07690 

08403 

09280 

10144 

11056 

12018 

13031 

13 

48 

07702 

08476 

09294 

101.59 

11072 

12034 

13048 

12 

49 

07715 

08489 

09308 

10174 

11087 

12051 

13065 

11 

50 

07727 

08503 

09323 

10189 

11103 

12067 

13083 

10 

51 

1.07740 

1.08516 

1.09337 

1.10204 

1.11119 

1.12084 

1.13100 

9 

52 

07752 

08529 

09351 

10218 

11134 

12100 

13117 

8 

53 

07765 

08542 

09365 

10233 

11150 

12117 

13135 

7 

54 

07778 

08556 

09379 

10248 

11166 

12133 

13152 

6 

55 

07790 

08569 

09393 

10263 

11181 

12150 

13170 

5 

56 

07803 

08582 

09407 

10278 

11197 

12166 

13187 

i 

57 

07816 

08596 

09421 

10293 

11213 

12183 

1.3205 

3 

58 

07828 

08609 

09435 

10308 

11229 

12199 

13222 

2 

59 

07841 

08623 

0!»449 

10323 

11244 

12216 

13240 

1 

60 

07853 

08636 

09464 

10338 

11260 

12233 

13257 

0 

/ 

68° 

67° 

66° 

65° 

64° 

63° 

62° 

/   1 

CO 

SECANTS. 

XI.-NATURAL  SECANTS 

AND  COSECANTS.  ' 

;ni 

/ 

SECANTS 

/ 

28° 

29" 

30° 

31° 

32° 

33° 

34° 

0 

1.13257 

1.14335 

1.15470 

1 . 1 6663 

1.17918 

1.192.36 

1.206V" 

U) 

1 

13275 

143.54 

15  ISO 

lt;US4 

17939 

1 92.-9 

201  4 

r9 

o 

13292 

H372 

15509 

16704 

1.7961 

19281 

20669 

5H 

3 

13310 

14391 

1.5528 

16725 

17982 

19304 

2(1693 

.^7 

4 

133i7 

14109 

15.548 

16745 

18004 

193J7 

2o;ir 

."6 

5 

13345 

14428 

15567 

16766 

18025 

19349 

20740 

"<.'i 

G 

13362 

14446 

1.5587 

16786 

18047 

19372 

20764 

51 

13380 

1 4465 

15(;06 

16806 

18068 

19394 

2(17  ^  8 

5i 

8 

13398 

14483 

15626 

16S27 

18090 

19417 

2(:8]2 

.-,•) 

9 

13415 

1  1502 

1.5645 

16848 

18111 

19440 

20Si6 

r.i' 

10 

13433 

14521 

1.5605 

16S68 

18133 

19463 

20859 

50 

11 

1.13451 

1.14539 

1.1 5684 

1.16889 

1.18155 

1.19485 

1.20883 

49 

1-J 

13468 

14558 

15704 

16909 

18176 

19508 

20907 

48 

13 

13486 

14.576 

15724 

16930 

18198 

19.531 

20931 

47 

14 

13504 

14595 

15743 

16950 

18220 

19.554 

20955 

46 

15 

13521 

14614 

1.5763 

16971 

18241 

19576 

20979 

45 

16 

13539 

14632 

15782 

16992 

18263 

19.599 

21003 

44 

17 

13557 

14651 

15802 

17012 

18285 

19622 

21027 

43 

18 

13575 

14670 

15822 

17033 

18307  " 

19645 

21051 

42 

19 

13593 

1 4689 

15841 

17054 

18328 

19668 

21075 

41 

20 

13610 

14707 

15861 

17075 

18350 

19691 

21099 

40 

21 

1.13628 

1.14726 

1.15881 

1.17095 

1.18372 

1.19713 

1.21123 

39 

22 

13646 

14745 

15901 

17116 

18394 

19736 

21147 

38 

23 

13G64 

14764 

15920 

17137 

18416 

19759 

21171 

37 

24 

13682 

14782 

15940 

171.58 

18437 

19782 

21195 

36 

2r) 

13700 

14801 

15960 

17178 

18459 

19805 

21220 

35 

26 

13718 

14820 

1.5980 

17199 

18481 

19828 

21244 

34 

27 

13735 

14839 

16000 

17220 

18503 

19851 

21268 

33 

28 

137.53 

14858 

16019 

17241 

18525 

19874 

21292 

32 

21) 

13771 

14877 

16039 

17262 

18547 

19897 

21316 

31 

30 

13789 

14896 

16059 

17283 

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31 

1.13807 

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1.16070 

1.17304 

1.18591 

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1.21365 

29 

32 

13825 

1 4933 

16099 

17325 

18613 

19967 

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28 

33 

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14952 

16119 

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18635 

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27 

34 

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20013 

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26 

35 

13879 

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161.59 

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20036 

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25 

36 

13897 

15009 

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18701 

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21487 

24 

37 

13916 

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13934 

1.5047 

16219 

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39 

139.52 

15066 

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18767 

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40 

13970 

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1.13988 

1.15105 

1.16279 

1.17514 

1.18812 

1.20176 

1.21609 

19 

42 

14006 

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16299 

17535 

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20199 

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18 

43 

14024 

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17 

44 

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45 

14061 

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163.59 

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46 

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14 

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49 

14134 

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16440 

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20363 

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11 

50 

141.52 

15277 

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17704 

19012 

20386 

21830 

10 

51 

1.14170 

1.1.5296 

1.16481 

1.17726 

1.19034 

1.20410 

1.21855 

9 

52 

14188 

1.5315 

16.501 

17747 

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8 

53 

14207 

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17768 

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1 

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1 4225 

15354 

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55 

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1.5393 

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/ 

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L.-    -   - 

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CHJSECANTS. 

312       XI.— NATURAL   SECANTS   AND   COSECANTS. 


/ 

SECANTS 

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1 
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35° 

36° 

37° 

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1.22077 

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1.22352 

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1.27221 

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1.32872 

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12 

22377 

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22402 

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22153 

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2.5711 

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20 

22579 

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1.22604 

1.24160 

1.25795 

1.27513 

1.29318 

1.31216 

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22 

22629 

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22680 

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22731 

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22833 

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1.22858 

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1.29628 

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l.:3:3554 

29 

32 

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22935 

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26 

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22960 

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3:3691 

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22986 

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24588 

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23089 

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41 

1.23114 

1.24696 

1.26358 

1.28105 

1.29940 

1.31870 

l.:33899 

19 

42 

23140 

24723 

26387 

28134 

29971 

31903 

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18 

43 

23166 

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30003 

31936 

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17 

44 

23192 

24777 

26443 

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30034 

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16 

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23217 

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15 

46 

23243 

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23269 

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26529 

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13 

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23295 

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30160 

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12 

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26586 

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32134 

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11 

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23347 

24910 

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10 

51 

1.23373 

1.24967 

1.26643 

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1.32201 

1.34247 

9 

52 

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8 

53 

23424 

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0 

/ 

54° 

53° 

52° 

51° 

50° 

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1 

1 

COSECANTS. 

f 

,. 1 

XL— NATURAL   SECANTS   AND   COSECANTS.       '61'6 


1 

SECANTS. 

• 

/ 

42° 

43° 

44° 

45° 

46° 

47° 

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0 

1.34503 

1.36733 

1.39016 

1.41421 

1.43956 

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1 

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9 

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37068 

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34917 

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1.34953 

1.37143 

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1.41876 

1.44435 

1.47134 

1.49981 

49 

12 

34988 

37180 

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41918 

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13 

35024 

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419.59 

44523 

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35060 

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15 

35095 

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45 

16 

35131 

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44 

17 

35167 

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18 

35203 

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20 

35274 

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40 

21 

1.35310 

1.37519 

1.39844 

1.42293 

1.44875 

1.47598 

1.50471 

39 

22 

35346 

37556 

39884 

42335 

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38 

23 

35382 

37594 

39924 

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37 

24 

35418 

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25 

35454 

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35490 

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40043 

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35526 

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31 

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35634 

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30 

31 

1.35670 

1.37898 

1.40243 

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29 

32 

35707 

37936 

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4.5631 

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1.36034 

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19 

42 

36070 

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4.5811 

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46 

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14 

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36253 

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43395 

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48 

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4.3438 

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49 

36327 

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40971 

43481 

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50 

36363 

38628 

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10 

51 

1.36400 

1.38666 

1.41053 

1.43567 

1.46218 

1.49015 

1.51968 

9 

52 

36437 

38705 

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53 

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7 

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36548 

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56 

36585 

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4 

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36622 

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58 

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59 

36<)96 

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41380 

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46628 

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0 

/ 

47° 

46° 

45° 

44" 

43° 

42° 

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/ 

COSECANTS. 

h514 

XL  NATURAL  SECANTS 

AND 

COSECANTS. 

0 

■ 

SECANTS. 

/ 

49° 

60° 

51° 

52° 

53° 

54° 

55° 

1.52425 

1.55572 

1.58902 

1.62427 

1.66164 

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1.74345 

60 

1 

52476 

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52579 

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4 

52630 

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52681 

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52^32 

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7 

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52835 

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52886 

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10 

52938 

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1.52989 

1.56169 

1.59533 

1.63096 

1.66873 

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1.75146 

49 

12 

53041 

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13 

53092 

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53196 

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53247 

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1.53507 

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1.71576 

1.75882 

39 

22 

53559 

56771 

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71646 

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38 

23 

53611 

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53663 

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53768 

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71925 

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53820 

57047 

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64081 

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53872 

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64144 

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72065 

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32 

29 

53924 

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31 

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53977 

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31 

1.54029 

1.57269 

1.6069S 

1.61330 

1.68183 

1.72275 

1.76626 

29 

32 

54082 

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68250 

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28 

33 

54134 

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34 

54187 

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54292 

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54345 

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77077 

23 

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20 

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1.54557 

1.57827 

1.61288 

1.64957 

1.68848 

1.72982 

1.77378 

19 

42 

54610 

57883 

61348 

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18 

43 

54663 

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17 

44 

54716 

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16 

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54769 

58051 

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65209 

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15 

46 

54822 

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61586 

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73338 

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47 

54876 

58164 

61646 

65336 

69250 

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13 

48 

54929 

58221 

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12 

49 

54982 

58277 

61765 

65462 

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11 

50 

55036 

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10 

51 

1.55089 

1.58390 

1.61885 

1.65589 

1.69520 

1.73696 

1.78138 

9 

52 

55143 

58447 

61945 

65653 

69587 

73768 

78215 

8 

53 

55196 

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65717 

69655 

73840 

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7 

54 

55250 

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62065 

657F0 

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73911 

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6 

55 

55303 

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62125 

65844 

69790 

73983 

78445 

5 

56 

55357 

58674 

62185 

65908 

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74056 

78521 

4 

57 

55411 

58731 

62246 

65972 

69926 

74128 

78598 

3 

58 

55465 

58788 

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66036 

69994 

74200 

78675 

2 

59 

55518 

58845 

62366 

66100 

70062 

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1 

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55572 

58902 

62427 

66164 

70130 

74345 

78829 

0 
/ 

/ 

40° 

39° 

38° 

37° 

36° 

35° 

34° 

i     - 

COSECANTS. 

XI.— NATURAL  SECANTS  AND  COSECANTS. 


315 


I 

SECANTS. 

/ 

56° 

57° 

58° 

69° 

60° 

61° 

62° 

0 

1.78859 

1.83608 

1.88708 

1.94160 

2.00000 

2.06267 

2  13005 

60 

1 

789C6 

83690 

88796 

94254 

00101 

00375 

13122 

59 

2 

78984 

83773 

88884 

94349 

00202 

00483 

1.32-39 

58 

3 

79061 

83855 

88972 

94443 

00303 

06592 

i;i356 

57 

4 

79138 

83938 

89060 

94537 

00404 

06701 

i;W73 

56 

5 

79216 

84020 

89148 

94632 

00505 

00809 

13590 

55 

6 

79293 

84103 

89237 

94726 

00007 

06918 

13707 

54 

t 

79371 

84186 

89325 

948-21 

00708 

07027 

13825 

53 

8 

79449 

84269 

89414 

94916 

00810 

07137 

13942 

52 

9 

79527 

84352 

89.503 

95011 

00912 

07246 

14060 

51 

10 

79604 

84435 

89591 

95106 

01014 

07356 

14178 

50 

n 

1.79682 

1.84518 

1.89680 

1.95201 

2.01116 

2.07465 

2.14296 

49 

i-j 

79701 

84601 

89709 

95296 

01218 

07575 

14414 

48 

13 

798:39 

84685 

89858 

95392 

01320 

07685 

14533 

47 

14 

79917 

84768 

89948 

9.5487 

01422 

07795 

14651 

46 

15 

79995 

84852 

90037 

95583 

01525 

07905 

14770 

45 

IC. 

80074 

8-1935 

90126 

95678 

01628 

08015 

14889 

44 

ir 

80152 

85019 

90216 

95774 

01730 

08126 

15008 

43 

18 

80231 

85103 

90305 

95870 

01833 

OS-230 

15127 

42 

I'J 

80309 

85187 

90395 

95966 

01930 

08347 

15246 

41 

L'O 

80388 

85271 

90485 

96062 

02039 

08458 

15366 

40 

'Jl 

1.80467 

1.8.5355 

1.90575 

1.96158 

2.0-2143 

2  08569 

2.15485 

39 

»•> 

80546 

85439 

90665 

96255 

02246 

08680 

15(i05 

•38 

•J3 

80025 

85523 

90755 

90351 

02349 

08791 

15725 

37 

•z\ 

80704 

85008 

90845 

90448 

02453 

08903 

15845 

36 

X!5 

80783 

85692 

90935 

90544 

02557 

09014 

15905 

35 

^'G 

80802 

85777 

91026 

90641 

02661 

09126 

16085 

34 

^  1 

80942 

85861 

91110 

96738 

02765 

09238 

16206 

33 

•JS 

81021 

85946 

91207 

96835 

02809 

09350 

16326 

32 

2".) 

81101 

80031 

91297 

90932 

02973 

0'.»462 

.  16447 

31 

30 

81180 

86116 

91388 

97029 

03077 

09574 

10568 

30 

1  31 

1.81260 

1 .86201 

1.91479 

1.97127 

2.03182 

2.09686 

2.16689 

29 

1  32 

81340 

80286 

91570 

972-24 

03286 

09799 

16810 

28 

I  38 

81419 

80371 

91661 

97322 

03391 

09911 

16932 

27 

34 

81499 

80457 

917.52 

974-20 

03496 

10024 

17053 

26 

35 

81.579 

80542 

91844 

97517 

03601 

10137 

17175 

25 

31J 

81059 

86627 

91935 

97015 . 

03706 

10250 

17-397 

24 

37 

81740 

86713 

920-.27 

97713 

03811 

10363 

17419 

23 

38 

81820 

80799 

92118 

97811 

03916 

10477 

17541 

22 

3'.) 

81900 

80885 

92210 

97910 

04022 

10590 

17663 

21 

40 

81981 

80990 

9230-2 

98008 

04128 

10704 

17786 

20 

41 

1.82061 

1.87056 

1.92394 

1.98107 

2.04233 

2.10817 

2.17909 

19 

42 

8-J142 

87142 

9-,'486 

98-205 

04339 

10931 

18031 

18 

43 

822-,>2 

87-229 

92578 

98304 

04445 

11045 

18154 

17 

44 

8-2303 

87315 

92670 

98403 

04551 

111.59 

18-277 

16 

45 

82384 

87401 

92762 

98502 

04658 

11274 

18401 

15 

40 

82465 

87488 

92855 

98601 

04764 

11388 

18524 

14 

47 

82546 

87574 

92947 

98700 

04870 

11503 

18648 

13 

48 

82627 

87661 

93040 

98799 

04977 

11617 

18772 

12 

49 

82709 

87748 

93133 

98899 

05084 

11732 

18895 

11 

50 

82790 

87834 

93:226 

98998 

05191 

11847 

19019 

10 

51 

1.8-2871 

1.87921 

1.93319 

1.99098 

2.05298 

2.11903 

2.19144 

9 

5',> 

809.53 

88008 

93412 

99198 

0.5405 

12078 

19268 

8 

53 

83034 

88095 

93505 

99298 

05512 

12193 

19393 

7 

54 

83116 

8S183 

93598 

99398 

0.5619 

12309 

19517 

6 

55 

83198 

88270 

9309-2 

99498 

05727 

124-25 

19642 

5 

50 

8;;2S0 

88357 

93785 

99598 

05835 

12.540 

19767 

4 

'•>  i 

83362 

88445 

93879 

9!I09S 

05942 

1-2657 

1 9892 

3 

58 

83444 

8S532 

93973 

99799 

00050 

12773 

20018 

o 

59 

83520 

88020 

94000 

99899 

00158 

12889 

20143 

1 

60 
1 
/ 

83008 

88708 

94100 

2.00000 

06207 

13005 

20269 

27° 

0 

33° 

32° 

81° 

30° 

29° 

28° 

/ 

COSECANTS. 

ue 

XI.     NATURAL   SECANTS 

AND 

COSEC. 

\NTS. 

/ 

SECANTS.                                                     1 

60 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

0 

2.20269 

2.28117 

2.36620 

2.45859 

2.559.30 

2.66947 

2.79043 

1 

20395 

28^:53 

36768 

46020 

56106 

67139 

792.54 

59 

2 

20521 

28390 

36916 

46181 

56282 

67332 

79466 

58 

3 

20647 

28526 

37064 

46342 

56458 

67525 

79079 

57 

4 

20773 

28663 

37212 

46504 

56634 

67718 

79891 

56 

5 

20900 

28800 

37361 

46665 

56811 

67911 

80104 

55 

6 

21026 

28937 

37509 

46S27 

56988 

68105 

80318 

51 

7 

21153 

29074 

376.58 

46989 

57165 

68299 

80531 

53 

8 

21280 

29211 

37808 

47152 

57342 

68494 

80746 

52 

9 

21407 

29349 

37957 

47314 

57520 

68689 

80960 

51 

10 

21535 

29487 

38107 

47477 

57698 

68884 

81175 

50 

11 

2.21662 

2.29625 

2.38256 

2.47640 

2.57876 

2.69079 

2.81390 

49 

12 

21790 

29763 

38406 

47804 

58054 

69275 

81605 

48 

13 

21918 

29901 

38556 

47967 

58233 

69471 

81821 

47 

14 

22045 

30040 

38707 

48131 

58412 

69667 

82037 

46 

15 

22174 

30179 

38857 

48295 

58591 

69864 

82254 

45 

16 

22302 

30318 

39008 

48459 

58771 

70061 

82471 

44 

17 

22430 

30457 

391.59 

48624 

58950 

70258 

82688 

43 

18 

22559 

30596 

39311 

48789 

59130 

70455 

82906 

42 

19 

22688 

.•i0735 

39402 

489,54 

59311 

70653 

83124 

41 

20 

22817 

30875 

39614 

49119 

59491 

70851 

83342 

40 

21 

2.22946 

2  31015 

2.39766 

2.49284 

2.. 5967  2 

2.710.50 

2.83561 

39 

' 

22 

23075 

31155 

39918 

49450 

59853 

71249 

83780 

38 

23 

23205 

31295 

40070 

49616 

60035 

71448 

83999 

37 

24 

23334 

31436 

40222 

49782 

60217 

71647 

84219 

36 

25 

23464 

31576 

40375 

49948 

60399 

71847 

84439 

35 

26 

23594 

31717 

40528 

.50115 

60581 

72047 

846.59 

34 

27 

23724 

31858 

40681 

502S2 

60763 

72247 

84880 

33 

28 

23855 

31999 

40835 

.50449 

60946 

72448 

85102 

32 

29 

23985 

32140 

409.^8 

.50617 

61129 

72649 

85323 

31 

30 

24116 

3228  i 

41112 

50784 

61313 

72850 

85.545 

30 

31 

2.24247 

2.32424 

2  41296 

2.509.52 

2.61496 

2.73052 

2.8,5767 

29 

32 

24378 

32566 

41450 

51120 

61680 

73254 

85990 

28 

33 

24509 

32708 

41(;05 

51289 

61864 

734.56 

80213 

27 

34 

24640 

32850 

41760 

51457 

62049 

73659 

86437 

26 

35 

24772 

32993 

41914 

51626 

62234 

73S62 

86661 

25 

36 

24903 

33135 

42070 

51795 

62419 

74065 

86885 

24 

37 

25035 

332^8 

42225 

51965 

62604 

74269 

87109 

23 

38 

25167 

33422 

423.S0 

52134 

62790 

74473 

87334 

22 

39 

25300 

33565 

42536 

52304 

62976 

74677 

87560 

21 

40 

25432 

33708 

42692 

52474 

63162 

74881 

87785 

20 

41 

2.25565 

2.33852 

2.42848 

2.52645 

2.63348 

2.750S6 

2.88011 

19 

42 

25697 

33996 

43005 

52815 

63.^>35 

75202 

88238 

18 

-1  ^ 

43 

25830 

34140 

43162 

52986 

63722 

75497 

88465 

1 1 

t  I* 

44 

25963 

34284 

43318 

53157 

63909 

75703 

88692 

16 

45 

26097 

34429 

43476 

53329 

64097 

75909 

8S9-J0 

15 
14 
13 

1  k) 

46 

26230 

34.573 

43633 

53500 

64285 

76116 

89148 

47 

26364 

34718 

43790 

5367-2 

64473 

76323 

8937'6 

48 

26498 

34863 

43948 

53845 

64662 

76530 

89605 

12 
11 
10 

49 

26632 

3.5009 

44106 

54017 

64851 

7C)737 

898:J4 

50 

26766 

35154 

44264 

54190 

65040 

70945 

90063 

51 

2.26900 

2.35.300 

2.44423 

2  54363 

2.6.5229 

2.771.54 

2.90293 

9 

8 

52 

27035 

3.5446 

44.582 

54536 

6.5419 

(  (  .502 

90524 

53 

27169 

35592 

44741 

54709 

65609 

77571 

90754 

1 

6 
5 
4 
3 

v> 

54 

27304 

35738 

44900 

54883 

65799 

77780 

90986 

55 

27439 

35885 

45059 

5.5057 

65989 

77990 

91217 

56 

27574 

36031 

4.5219 

.55231 

66180 

78200 

91449 

57 

27710 

36178 

45378 

55405 

6(1371 

78410 

91681 

58 

27845 

36325 

45539 

5.5580 

66563 

7S021 

91914 

1 
0 

59 

27981 

36473 

45699 

55755 

667.55 

78832 

92147 

60 

28117 

36620 

458.59 

55930 

66947 

79043 

92380 

J 

/ 

26° 

25° 

24° 

23°           22° 

OSECANTS. 

21° 

20° 

' 

C< 

XI.— NATURAL   SECANTS   AND   COSECANTS. 


317 


/ 

SECANTS. 

/ 

70° 

71° 

7-2° 

73° 

74° 

75° 

76° 

0 

2.92380 

3.07155 

3.23607 

3.42030 

3.62796 

3.86370 

4.13357 

60 

1 

92614 

07415 

23897 

42356 

63164 

86790 

13839 

59 

a 

92849 

07675 

24187 

42683 

63533 

8721 1 

14323 

58 

3 

93083 

07936 

24478 

43010 

6i90:] 

87633 

14809 

57 

4 

93318 

08197 

24770 

43337 

64274 

88056 

15295 

56 

5 

93554 

08459 

25062 

43666 

64645 

88179 

15782 

55 

6 

93790 

08721 

25355 

439!»5 

65018 

88904 

16271 

54 

7 

94026 

08983 

25648 

44324 

65391 

89330 

16761 

53 

8 

94263 

09246 

25942 

44655 

65765 

89756 

172.52 

52 

9 

94500 

09510 

26237 

44986 

66]40 

90184 

17744 

51 

10 

94737 

09774 

26531 

45317 

66515 

90613 

18238 

50 

11 

2.94975 

3.10038 

3.26827 

3.45650 

3.66892 

3.91042 

4.18733 

49 

12 

95213 

10303 

27123 

45983 

67269 

91473 

19228 

48 

18 

95452 

10568 

27420 

46316 

67647 

91904 

19725 

47 

14 

95691 

10834 

27717 

46651 

68025 

92337 

20224 

46 

15 

95931 

11101 

28015 

46986 

68405 

92770 

20723 

45 

16 

96171 

11367 

28313 

47321 

68785 

93204 

21224 

44 

17 

96411 

11635 

28612 

47658 

69107 

93640 

21726 

43 

18 

96652 

11903 

28912 

47995 

69549 

91076 

22229 

42 

19 

96^93 

12171 

29212 

48333 

69931 

94514 

22734 

41 

20 

97135 

12440 

29512 

48671 

70315 

94952 

23239 

40 

21 

2.97377 

3.12709 

3.29814 

3.49010 

3.70700 

3.95392 

4.23746 

39 

22 

97619 

12979 

30115 

49350 

71085 

95832 

242.55 

38 

28 

97862 

13249 

30418 

49691 

71471 

96274 

24764 

37 

24 

98106 

13520 

30721 

50032 

71858 

96716 

2.5275 

36 

25 

98349 

13791 

31024 

50374 

72246 

97160 

25787 

35 

26 

9S594 

14063 

31328 

50716 

72635 

97604 

26300 

34 

27 

98838 

14335 

31633 

51060 

73024 

98050 

26814 

33 

28 

99083 

14608 

31939 

51404 

73414 

98497 

27330 

32 

29 

993-.'9 

14881 

32244 

51748 

73806 

98944 

27847 

31 

30 

99574 

15155 

32551 

52094 

74198 

993f)3 

28366 

30 

31 

2.99821 

3.15429 

3.328.58 

3.52440 

3,74591 

3.99843 

4.288S5 

29 

32 

3.00067 

15704 

33166 

52787 

74984 

4.00293 

29406 

28 

33 

00315 

15979 

33474 

53134 

75379 

00745 

29929 

27 

34 

00562 

16255 

33783 

53482 

75775 

01198 

30452 

26 

35 

00810 

16531 

34092 

53831 

76171 

01052 

30977 

25 

36 

01059 

16808 

34403 

54181 

76.568 

02107 

31503 

24 

3V 

01308 

17085 

34713 

54531 

76966 

02563 

32031 

23 

38 

01557 

17363 

35025 

54883 

77365 

03020 

32560 

22 

39 

01807 

17641 

35336 

55235 

77765 

03479 

33090 

21 

40 

02057 

17920 

35649 

.55587 

78166 

03938 

33622 

20 

41 

3.02308 

3.18199 

3.35962 

3.55940 

3.78568 

4.04398 

4.34154 

19 

42 

02559 

18479 

36276 

56294 

78970 

04800 

34689 

18 

43 

02810 

18759 

36590 

56649 

79374 

05322 

35224 

17 

44 

03062 

19040 

30905 

57005 

79778 

05786 

35761 

16 

45 

03315 

19322 

37221 

57361 

80183 

06251 

36299 

15 

46 

03568 

19604 

37537 

57718 

80589 

06717 

36839 

14 

47 

03821 

19886 

37854 

58076 

80996 

07184 

37380 

13 

48 

04075 

20169 

38171 

58434 

81404 

07652 

37923 

12 

49 

04329 

20453 

38489 

58794 

81813 

08121 

38466 

11 

50 

04584 

20737 

38808 

59154 

82223 

08591 

39012 

10 

51 

3.04839 

3.21021 

3.39128 

3.59514 

3.82633 

4.09063 

4.395.58 

9 

52 

05094 

21306 

39448 

59876 

83045 

09535 

40106 

8 

53 

05350 

21592 

39768 

60238 

83457 

10009 

40656 

7 

54 

05607 

21878 

40089 

60601 

83871 

10484 

41206 

6 

55 

05864 

22165 

40411 

60965 

84285 

10960 

41759 

5 

56 

00121 

22452 

40734 

61330 

84700 

11437 

42312 

4 

57 

06379 

22740 

41057 

61695 

8.5116 

11915 

42867 

3 

58 

06637 

23028 

41381 

62061 

85533 

12394 

43424 

2 

59 

06896 

23317 

41705 

62428 

85951 

12875 

43982 

1 

60 

07155 

23607 

42030 

62796 

86370 

13357 

44541 

0 

/ 

19" 

18° 

17° 

16" 

16° 

14° 

13° 

/ 

COSECANTS. 

318       XL— NATURAL  SECANTS  AND  COSECANTS. 


/ 

SECANTS. 

/ 

77° 

78° 

79° 

80° 

81° 

82° 

83° 

0 

4.44541 

4.80973 

5.24084 

5. (Obi  i 

6.39245 

7.18530 

8.205.51 

60 

1 

45102 

81633 

24870 

76829 

40422 

20020 

22500 

59 

2 

45664 

82294 

25658 

77784 

41602 

21517 

244.57 

58 

3 

46228 

82956 

26448 

78742 

42787 

23019 

26425 

57 

4 

46793 

83621 

27241 

79703 

43977 

24529 

28402 

56 

5 

47360 

84288 

28036 

80667 

45171 

26044 

.30388 

55 

6 

479r^8 

84956 

28833 

81635 

46369 

27566 

321384 

54 

1 

48498 

85627 

29634 

82606 

47572 

29095 

34390 

53 

8 

49069 

86299 

30436 

83581 

48779 

30630 

36405 

52 

9 

49642 

86973 

31241 

84558 

49991 

32171 

38431 

51 

10 

50216 

87649 

32049 

85.539 

51208 

33719 

40466 

50 

11 

4.50791 

4.88327 

5.32859 

5.86524 

6.52429 

7.35274 

8.42511 

49 

12 

51368 

89(X>7 

33t571 

87511 

53655 

36835 

44566 

48 

13 

51947 

89689 

34486 

88502 

54886 

38403 

46632 

47 

14 

52527 

90373 

35304 

89497 

56121 

39978 

48707 

46 

15 

53109 

91058 

36124 

90495 

57361 

41560 

50793 

45 

16 

53692 

91746 

36947 

91496 

58606 

43148 

52889 

44 

17 

54277 

92436 

37772 

92501 

598.55 

44743 

54996 

43 

18 

54863 

93128 

38000 

93.509 

61110 

46346 

.57113 

42 

19 

55451 

9.3821 

39430 

94.521 

62369 

479.55 

59241 

41 

20 

56041 

94517 

40263 

95536 

63'J33 

49571 

61379 

40 

21 

4.56632 

4.95215 

5.41099 

5.965.55 

6.64902 

7.51194 

8.6.3.528 

39 

22 

57224 

95914 

41937 

97.577 

66176 

52825 

65688 

38 

23 

57819 

96616 

42778 

98603 

6^4.54 

54462 

67859 

37 

24 

58414 

97320 

43622 

99633 

68738 

56107 

70041 

36 

25 

59012 

98025 

444r,8 

6.00666 

70027 

57759 

72234 

35 

26 

59611 

98733 

45317 

01703 

71321 

59418 

74438 

34 

27 

60211 

99443 

46169 

02743 

72620 

61085 

76653 

33 

28 

60813 

5.00155 

47023 

03787 

73924 

62759 

78880 

32 

29 

61417 

00869 

47881 

04834 

75233 

64441 

81118 

31 

30 

62023 

01.585 

48740 

05886 

76547 

66130 

83367 

30 

31 

4  62630 

5.02303 

5.49603 

6.06941 

6.77866 

7.67826 

8.8.5628 

29 

32 

63238 

03024 

50468 

08000 

79191 

69530 

87901 

28 

33 

63849 

03746 

51337 

09062 

80521 

71242 

90186 

27 

34 

64461 

04471 

52208 

10129 

818.56 

72962 

92482 

26 

35 

65074 

05197 

53081 

11199 

83196 

74689 

94791 

25 

36 

65690 

05926 

53958 

12273 

84542 

76424 

97111 

24 

37 

66307 

06657 

54837 

1.3350 

8.5893 

78167 

99444 

23 

38 

06925 

0:.39C 

55720 

14432 

87250 

79918 

9.01788 

22 

39 

67545 

08125 

56605 

1.5517 

88612 

81677 

04146 

21 

40 

68167 

0S863 

57493 

16607 

89979 

83443 

06515 

20 

41 

4.68791 

5.09602 

5.5S383 

6.17700 

6.913.52 

7.85218 

9.08897 

19 

42 

69417 

10344 

59277 

18797 

92731 

87001 

11292 

18 

43 

70044 

11088 

60174 

19898 

94115 

88792 

13699 

17 

44 

70673 

11835 

01073 

21004 

95505 

90592 

16120 

16 

45 

71303 

12.58? 

61976 

22113 

96900 

92400 

18553 

15 

46 

71935 

13334 

62881 

23226 

98301 

94216 

20999 

14 

47 

72569 

14087 

63790 

24343 

99708 

96040 

2.3459 

13 

48 

73205 

14842 

64701 

25464 

7.01120 

97873 

25931 

12 

49 

73843 

15.599 

6.5616 

26590 

025.38 

99714 

28417 

11 

50 

74482 

16359 

66533 

27719 

03962 

8.01565 

.30917 

10 

51 

4.7.5123 

5.1:121 

5.67454 

6.288.53 

7.0.5392 

8.03423 

9.33430 

9 

52 

75766 

17880 

68377 

29991 

06828 

05291 

35957 

8 

53 

76411 

18652 

69304 

31133 

08269 

07167 

.38497 

54 

77057 

19421 

70234 

32279 

09717 

09052 

410.52 

'6 

55 

77705 

20193 

71166 

33429 

11171 

10946 

43620 

5 

56 

783.55 

20966 

72102 

34.584 

12630 

12849 

46203 

4 

57 

79007 

21742 

73041 

35743 

14096 

14760 

48800 

3 

58 

79661 

22521 

73983 

36906 

1.5508 

16681 

51411 

2 

59 

80316 

23301 

74929 

380^3 

17046 

18612 

54037 

1 

60 

80973 

24084 

75877 

39245 

18530 

20551 

56677 

0 

/ 

12° 

11° 

10° 

9° 

8° 

7° 

6° 

J 

COSECANTS. 

XI.— NATURAL  SF.CANTS  AND  COSECANTS. 


319 


/ 

SECANTS. 

/ 

84^ 

8.5° 

86° 

87° 

88° 

89° 

0 

9.56677 

11.47.371 

14.3.3.5.59 

19.10732 

28.6.5371 

57.29869 

60 

1 

59332 

51199 

39.547 

21397 

89440 

58.26976 

59 

o 

62002 

55052 

455S6 

32182 

29.13917 

59.27431 

58 

3 

64687 

58932 

51676 

43088 

38812 

60.31411 

57 

4 

67387 

62837 

.57817 

54119 

64137 

61.39105 

56 

5 

70103 

06769 

61011 

65275 

89903 

62.. 5071 5 

55 

6 

72833 

70728 

70258 

76.560 

30.16120 

63.66460 

54 

7 

75579 

74714 

76558 

87976 

42802 

04.86.572 

53 

8 

78341 

78727 

82913 

99524 

69960 

66.11304 

52 

9 

81119 

82768 

89323 

20.11208 

97607 

67.40927 

51 

10 

83912 

86837 

95788 

23028 

31 .25758 

68.75730 

TjO 

11 

9.80722 

11.90934 

15.02310 

20.34989 

31.. 54425 

70.16047 

49 

12 

89547 

95060 

08890 

47093 

83023 

71.62285 

48 

13 

92389 

99214 

1.5.527 

.59341 

32.13.366 

73.14.583 

47 

14 

95248 

12.03.397 

22223 

71737 

43671 

74.73586 

46 

15 

98123 

07610 

28979 

84283 

74554 

76.396.55 

45 

16 

10.01015 

11852 

35795 

96982 

33.06030 

78.1.3274 

44 

17 

03923 

16125 

42072 

21.09838 

38118 

79.94968 

43 

18 

00849 

20427 

49611 

22852 

70835 

81.85315 

42 

19 

09792 

24761 

■  56614 

36027 

34.04199 

83.84947 

41 

20 

12752 

29125 

63679 

49368 

38232 

85.94561 

40 

21 

10.1.5730 

12.33.521 

15.70810 

21.62876 

34.72952 

88.14924 

39 

22 

18725 

37948 

78005 

76555 

35.08380 

90.46S86 

38 

23 

21739 

42408 

85268 

90409 

44.539 

92.91387 

37 

24 

24770 

46900 

92597 

22.04440 

81452 

95.49471 

36 

2?^ 

27819 

51424 

99995 

18653 

36.19141 

98.22303 

35 

26 

30887 

55982 

16.07462 

33050 

57633 

101.11185 

34 

27 

33973 

60572 

14999 

47635 

96953 

104.17574 

33 

28 

37077 

65197 

22607 

62413 

37.37127 

107.43114 

32 

29 

40201 

69856 

30287 

77386 

78185 

110.89656 

31 

30 

43343 

74550 

38041 

92.559 

38.20155 

114.59301 

30 

31 

10.46505 

12.79278 

16.4.5869 

23.07935 

38.63068 

11 8,. 54440 

29 

3i 

49685 

84042 

53772 

23520 

39.06957 

122.77803 

28 

33 

52886 

88841 

61751 

39316 

51855 

127.32526 

27 

34 

56106 

93677 

69808 

55329 

97797 

1.32.22229 

26 

35 

59346 

98549 

77944 

71503 

40.44820 

137.51108 

25 

3(; 

62605 

13.034.58 

86159 

88022 

92963 

143.24061 

24 

37 

65S85 

08040 

944.56 

24.04712 

41.42266 

149.46837 

23 

38 

69186 

13388 

17.02835 

21637 

92772 

156.26228 

22 

3'J 

72507 

18411 

11297 

38802 

42.44525 

163.70325 

21 

40 

75S49 

23472 

19843 

56212 

97571 

171.88831 

20 

41 

10.79212 

13.28572 

17.28476 

24.7.3873 

43.51961 

180.93496 

19 

42 

82.596 

3.3712 

.37196 

91790 

44.07746 

190.98680 

18 

43 

80001 

38891 

40005 

25.09969 

64980 

202.22122 

17 

44 

89428 

44112 

51903 

28414 

45.23720 

214.8.5995 

16 

4'> 

92877 

49373 

63893 

47134 

84026 

229.18385 

15 

4C> 

96:348 

54076 

72975 

66132 

46.4.5963 

245.55402 

14 

47 

99841 

60021 

82152 

85417 

47.09.596 

264.44269 

13 

48 

11.033.^6 

65408 

91424 

26.04994 

74997 

286.47948 

12 

49 

06894 

70838 

18.00794 

24869 

48.42241 

312.52297 

11 

50 

10455 

76312 

10262 

45051 

49.11406 

343.77516 

10 

51 

11.14039 

13.«1829 

18.19830 

26.65546 

49.82.576 

381.972.30 

9 

52 

17646 

87391 

29501 

86360 

50.. 5.5840 

429.71873 

8 

r)3 

21277 

92999 

39274 

27.07.503 

51 .31290 

491.10702 

7 

54 

24932 

98651 

49153 

28981 

52.09027 

572.9.5809 

6 

55 

28610 

14  04350 

.59139 

.50804 

891.56 

687.54960 

5 

50 

32313 

10096 

69233 

72978 

.53.71790 

8.59.43689 

4 

57 

36040 

1.5889 

79438 

9.5513 

54.57046 

1145.9157 

3 

58 

397!»2 

21730 

897.55 

28.18417 

.55.4.^^053 

1718.8735 

2 

59 

43569 

27620 

19.00185 

41700 

56.35946 

3437.7468 

1 

60 

47371 

33559 

10732 

65371 

57.29869 

00 

0 

1 

5° 

4° 

3° 

2° 

1° 

0° 

/ 

COSECANTS. 

320 


TABLE  XU.— TANdENTS   AND  COTANGENTS. 


"o 

0»            1 

1             1°             ! 

2°            1 

3 

° 

/ 

60  . 

Tang 

.00000 

Cotang 

Tang 

.01746 

Cotang 

Tang 

.0.3492 

Cotang 

Tang  ! 
.05241 

Cotang 

Infinite. 

57.2900 

28.6363 

19.0811 

1 

.00029 

3437.75 

.01775 

56.3506 

.0.3.521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711  1 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9.372 

.05328 

18.7678 

57 

'  4 

.00116 

859.436 

.01862 

53.7036 

.03609 

27.7117 

.053.57 

18.6656 

56 

5 

.00145 

6S7.549 

.01891 

52.0821 

.03638 

27.4899 

.05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.0.5416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.. 30.32  : 

.0.3696 

27.0566 

.05445 

18.3655 

.53 

8 

.00233 

429.713 

.01978 

50.5485  : 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.81.^7  1 

.03754 

26.6.367 

.05503 

18.1708 

51 

10 

.00291 

a43.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.0.3312 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05501 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8;M8 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.0.3929 

25.4517 

.05678 

17.6106 

45 

IG 

.004G5 

214.858 

.022U 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

2U2.219 

.02240 

44.6386 

.0.3987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.. 5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

34.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4.335 

.04104 

24.3675 

.05854 

17.08.37 

39 

22 

.00040 

156.259 

.02386 

41.9158 

.04133 

24.19.57 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4.358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

,02502 

39.9655 

.04250 

23.5.321 

.05999 

16.6681 

M 

27 

.00785 

127.321 

.02531 

39.50.59 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0.568 

.04308 

23.21,37 

.06058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

.38.6177 

.04337 

23.0.577 

.06087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.^99 

30 

31 

.00902 

110.892 

.02648 

.37.7688 

.04.395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.. 3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9.560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.. 5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

,02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

.35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

;35. 4.313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

S4.3678 

.04658 

21.4704 

.OfrlOS 

15.6048 

20 

41 

.01193 

as.  84.35 

.029.39 

34.0273 

.04687 

21.3.369 

.064.37 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.. 3662 

,04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.04.52 

.04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7.3<;3 

.04803 

20.8188 

.06.554 

15.2.571 

15 

46 

.C1338 

74.7292 

.030^1 

32.4..,13 

.048.33 

20.69.32 

.06584 

15.1893 

14 

47 

.01367 

73.1.390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0.557 

12 

49 

.01425 

70.1.5.33 

.03172 

31.. 5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.067.30 

14.8,596 

9 

52 

.01513 

66.1055 

.a3259 

30.6833 

.05007 

19.9702 

.06759 

14.79.54 

8 

53 

.01542 

64.8580 

.0.3288 

30.4116 

.0.5037 

19.8546 

.06788 

14.7.317 

i 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

C 

55 

.01608 

62.4992 

.0.3.346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03.376 

29.6245 

.0.5124 

19.5156 

.06876 

14.54.38 

4 

57 

.01658 

60.3058 

.0.3405 

29.3711 

.0.51.53 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.0^434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

o 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.. 3607 

1 

60 

/ 

.-■■1   ■! 

.01746 
Cotang 

57.2900 

.03492 

28.6363 

.05241 
Cotang 

19.0811 

.06993 
Cotang 

14.3007 
Tang 

0 

f 

Tang  ' 

Cotang 

Tang 

Tang 

8 

9° 

88° 

i           87» 

8 

6° 

TABLE   XII.-TAiSraENTS   AND   COTANGENTS. 


321 


40                1 

5°     ■        1 

6^ 

•J 

'0 

60 

/ 

Tang 
.0(5993 

Cotang 

Tang 
.08749 

Cotang 
11.4:301 

Tang 
.10510 

Cotang 

Tang 
.12278 

Cotang 
8.14435 

T) 

14.3007 

9.51436 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3^0 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.5W17 

.10657 

9.38307 

.12426 

8.04756 

55 

6 

.07168 

13.9507  I 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9882 

.10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.56:^4 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

•45 

IG 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07ol9 

13.2996 

1    .09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7157 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5402  i 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

33 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10. 4172 

.11364 

8.79964 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.61;24 

.09088 

10.3224  ■ 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

38 

.08104 

12.3390 

.09804 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.09893 

10.1080 

.IK    ' 

8.57718 

.13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.0995?^ 

10.0483 

.11718 

P.  53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11800 

8.47007 

.13580 

7.36389 

16 

45. 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08389 

11.9923 

.10099 

9.90211 

.11805 

8.42795 

.13639 

7.33190 

14 

^•; 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

iS 

.08397 

11.9087 

.10158 

9.ai482 

.11924 

8.38625 

.13698 

7.30018. 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442- 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.0a544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10.393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11.5072 

.10452 

9.. 56791   1 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 
Cotang 

11.4301 
Taug 

j   .10510 
Cotang 

8 

9.51436  I 
Tang    1 

40             1 

.12278 
Cotang 

8.14435 

!    .14054 
Cotang 

7.11537 

0 

1 

/ 

Tang 

Tang 

8 

5° 

8 

3° 

8 

2° 

622 


TA&uE  XII.-iANUENTS  AND  COTANGENTS. 


/ 

8° 

!             9° 

10° 

11° 

/ 

Tang 

j  Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

1  Cotang 

~0 

.14054 

i  7.11537 

.158^38 

1  6.31375 

.176:33 

5.67128 

.194:38 

5.14455 

60 

1 

.14084 

7.10038 

.15S68 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.08.546 

. 15898 

6.29007 

.17693 

5.6.5205 

.19498 

i  5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19.529 

i  5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.195.59 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.2.>i86 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24.321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

i   .16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

0.20851 

.17903 

5.58573 

.19710- 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.9.5385 

.16167 

6.18559  ' 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

'.14410 

6.93952 

'   .16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

0.16283 

.18023 

5.54851 

.19831 

5.042G7 

47 

14 

.14470 

6.91104 

.162.56 

6.1.5151  ( 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

0.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779  , 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

1   .16376 

6.10064  i 

.18173 

5.50264 

.1^982 

5.00451 

42 

19 

.14618 

6.84082 

i   .16405 

0.09552  , 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

1 

0.08444  1 

.18233 

5.48451 

.20012 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.7W564 

.16525 

6.05143 

.18:323 

5.4.5751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

. 16555 

6.04051  j 

.18:3.53 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.75838 

;   .16585 

6.02902 

.18:384 

5.4:3966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.1^856 

6.73133 

.16645 

6.0079'? 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.166;-4 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

02 

29 

.14915 

6.70450 

.16704 

5. 9% 46  ' 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

0.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96.510 

.18.564 

5.38677 

.20376 

4.90785 

29 

32 

.  15005 

6.66463 

.16794 

5.95448 

.18.594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

26 

35 

.1.J094 

6.62523 

.163S4 

5.92283  : 

.18684 

5.35206 

.20497 

4.87882 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34S45 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.5S627 

.16974 

5.89151 

. 18775 

5.326:31 

.20.588 

4.85727 

22 

39 

.15213 

6.. 57339 

.17004 

5. 881 14 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.1.5272 

6.54777 

.17063 

5.86a51 

.18865 

5.. 30080 

.20679 

4.83590 

19 

42 

.15302 

6.53.503 

.17093 

5.85024  ' 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15:332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.. 50970 

.17153 

5.82982 

.189.55 

5.275.53 

.20770 

4.81471 

16 

45 

.  15391 

'6.49710 

.17183 

5.81966 

.18986 

5.26715 

,   .20800 

4.80769 

15 

46 

.15421 

6.48456  1 

.17213 

5.80953 

.19016 

5.25.880 

.   .20830 

4.80068 

14 

47 

.15451 

6.47206  ; 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.4.5961  1 

;   .17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720  1 

!   .17303 

5.77936 

.19106 

5.23:391 

.20921 

4.77978 

11 

50 

.15540 

6.43484  i 

1   .17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

51 

.15570 

6.42253 

i   .17363 

5.7.5941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

!   .17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

!   .17423 

5.73960 

.19227 

5.2<1107 

.21043 

4.75iil9 

7 

54 

.15660 

6.38587 

.174.53 

5.72974 

.192.57 

5.19293 

.21073 

4.74.534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.. 361 65 

.17513 

5.71013 

.19:317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.. 34961 

.17.54:3 

5.700:37  j 

.19:347 

5.16863 

.21164 

4.72490 

3 

TjS 

.15779 

6.33761 

.17573 

5.69064 

.19:378 

5.16058 

.21195 

4.71813 

2 

:>9 

.1.5809 

6.32.566 

.17603 

5.68094 

.19408 

5.152.56 

.21225 

4.711:37 

1 

CO 

.15838 

6.31375  1 

.17633 
Cotang 

8 

5.67128  , 

.194:38 
Cotang 

5.1+4.55 

.21256 
Cotang 

4,70463 

0 

Cotang 

Tang 

Tang    1 
0°           ' 

Tang 

Tang 

81°             ' 

79° 

7 

8° 

TABLE  XIL— TANGENTS  AND  (JUTaNUENTS. 


:333 


/ 

"o 

12°           1 

13°           1 

14°           1 

15°             1 

/ 
60 

-Tang 

.21256 

Cotang 

Tang 
.23087 

Cotang 

Tang 
.249;^ 

Cotang  1 

Tang 
.26795 

Cotang 

4.70463 

4.33148 

4.01078 

3.73205 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.tK)582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72X38 

58 

3 

.21347 

4.68452 

.23179 

4.31430  i 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.215^9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.2.5335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219  1 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45  1 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.254.59 

3.92793 

.27326 

3.65957 

43 

■i8 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.5800f 

.23670 

4.22481 

.25.521 

3.91839 

.27388 

3.6.5121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

22 

.219C5 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.5.5458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756  : 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.52017 

4.54196 

.23^54 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53.568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22109 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3  60588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.2.5924 

3.85745 

.27795 

3.59775 

28^ 

33 

.22201 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.278.58 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

36 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

20079 

3.83449 

.27952 

3.57758 

38 

.224U 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

v.\ 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.b6557 

20 

41 

.22505 

4.44338 

,24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

t2 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.. 55761 

18 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.553G4 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

16 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

4G 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.. 54179 

14 

47 

.22GS9 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.401.52 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.. 53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.264.52 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.32872 

4.37207 

.24717 

4.04.586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

S . 75388 

.28517 

3.50666 

5 

56 

.22964 

4.354.59 

:   .24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.a4879 

!   .24840 

4.02.574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.. 3^1300 

'    .24871 

4.02074 

.26733 

3.74075 

.28612 

3.49.509 

2 

5!) 

.23056 

4.33723 

.21902 

4.01.576 

.2()764 

3.73640 

.28643 

3.49125 

1 

60 
/ 

.2;i087 
Cotang 

4.33148 

.249:i3 
Cotang 

4.01078 

.26795 
Cotang 

3.73205 

.28675 
,  Cotang 

3.48741 

0 

Tang 

Tang 

Tang 

Tang 

77° 

!          76° 

75° 

7 

4° 

324 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


"o 

16°           1 

17°           1 

i            18°            1 

19° 

/ 
60 

Tang 

.28675 

Cotang 

Tang 
..30573 

Cotang 

Tang 
.32492 

Cotang 

Tang 
.34433 

Cotang 

3.48741 

3.27085 

3.07768 

2.90421 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

1   .32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

i   .32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

i   .32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080  1 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

.53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

•   9 

.28958 

3.45.327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2  87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.874.30 

h9 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.041.52 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.0.3854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

1.5 

.29147 

3.4.3084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.8.5822 

43 

18 

.29242 

3.41973  1 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604  i 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40809 

.31242 

3.20079 

.33169 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

..31274 

3.19752 

.3.3201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.3.3233 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.33266 

3.00611 

.35216 

2.8.3965 

36 

25 

.29463 

3.39406  1 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.. 39042 

.31402 

3.18451 

.33330 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38079 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.8,3176 

33 

28 

.29.558 

3.. 3831 7 

.31466 

3.17804 

.33395 

2.99447 

.35.346 

2.82914 

32 

29 

.29590 

3.379.55 

.31498 

3.17481 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.296.53 

3.372.34 

.31.562 

3.16838 

.33492 

2.98.580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

.o3524 

2.98292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.3.3557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3. 361.58 

.31658 

3.15877 

.3.3589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.. 3.5800 

.31690 

3.1.55.58  1 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35608 

2.80833 

24 

37 

.29843 

3.35087 

.317.54 

•  3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.3.3718 

2.90573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.96288  1 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31883 

3.13656 

..33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.3.5805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.3.3881 

2.9.5155  i 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.3.3913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.. 32264 

.32010 

3.12400 

.a3945 

2.94591 

.35904 

2.78523 

15 

46 

..30128 

3.. 31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31.565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30193 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3,. 30868 

.32139 

3.11153 

.34075 

2.9.3468 

.36035 

2.^7507 

11 

50 

.30255 

3.30521 

.32171 

3.10843 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.767.50 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32.331 

3.09298  ! 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32:363 

3.08991 

.34303 

2.91.523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32.396 

3.08685 

.343.35 

2.91246 

.36298 

2.7.5496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34.368 

2.90971 

.363;31 

2.75246 

2 

59 

.30541 

3.27426 

..32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 

/ 

.30573 
Cotang 

3.270R5 

.32492 

3.07768 

.34433 
Cotang 

2.90421 

.36397 
Cotang 

2.74748 

0 

Tang 

Cotang 

Tang 

Tang 

Tang 

73° 

■1 
72°            1 

7 

1° 

70° 

TABLE  XII.— ^ 

rANGENI 

rS   AND 

COTANGENTS. 

"o 

20° 

21" 

22° 

1            23'^ 

60 

Tang 

.36397 

Cotang 
2.74748 

Tang 
.38386 

Cotang 

2.00509 

Tang 
.40403 

Cotang 
2.47509 

1  Tang 

Cotang 

.42447 

2.35585 

1 

.36430 

2.74499 

.38420 

2.60283 

-.40436 

2.47302 

.42482 

2.35395 

59 

2 

]    .36463 

2.74251 

.384.53 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2..59S31 

.40504 

2.46888 

.42551 

2.35015 

57 

4 

.36529 

2.73756 

.38520 

2.59006 

.40538 

2.46682 

.42585 

2.^4825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.^4636 

55 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7 

.36628 

2.73017 

.38620 

2.589.32 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

'   .36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36691 

2.72526 

.38687 

2.58484 

.40707 

2.4.5655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.4&451 

.42791 

2.33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

40 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56487 

.41018 

2.43825 

.43067 

2.. 321 97 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43023 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.4*422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

30 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2  07937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41020 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43510 

2.29801 

29 

32 

.37455 

2.66989 

.39458 

2.53433 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.395.59 

2.52786 

.41592 

2.40432 

.43054 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.437o8 

2.28528 

22 

39 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831. 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.. 51076 

.41865 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

41899 

2.38668 

.43066 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.26909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

,44105 

2.26730 

12 

49 

.38030 

2.G3021 

.40031 

2.49807 

.42070 

2,37697 

.44140 

2.20.552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

..38120 

2.62332 

.40132 

-2.40177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42-307 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36?3;3 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.422^6 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2  48340 

.42310 

2.36349 

.44:384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

..44418 

2.2.5132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.3.5967 

.44453 

«. 24956 

2 

59 

.38353 

2.60736 

.40369  , 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

60 

/ 

.;38386 

Cotang 

2.60509 

.40403  1 
Cotang ! 

2.47509 
Tang    1 

.42447 
Cotang 

2.3.5.585 

.^523 

2.24604 

0 

Tang 

Tang 

Cotang :    Tang 

69» 

6 

8<^ 

67» 

66° 

326 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


~0 

24°            1 

25° 

26° 

27° 

60 

Tang 
.44523 

Cotang 

Tang 
.46631 

Cotang 

'   Tang 
.48773 

Cotang  ' 
2.05U30 

Tang 

.50953 

Cotang 

2.24604 

2.14451 

1.96261 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

>    .48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44062 

2.23902 

.46772 

2.1:3801 

.48917 

2.04426 

.51099 

1.95698  '56  1 

5 

.44097 

2.23727 

.46808 

2.1:3039 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44733 

2.23553 

.46.S43 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.2:3378 

.46879 

2.1.3:316 

.49026 

2.03975 

.51209 

1.95277 

53 

8 

.44802 

2.23204 

.46914 

2.1:3154 

.49063 

2.0:3825 

.51246 

1.95137 

52 

9 

.44837 

2.2.30.30 

.46950 

2.12993 

I    .49098 

2.0.3(575 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

,46985 

2.12833 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

!    .49170 

2.03.376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.223.37 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.43012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94.301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631  ! 

.51540 

1.94023 

44 

17 

.45117 

2.21647 

.472:34 

2.11711 

.49387 

2.0248:3 

.51o<  i 

1.93885 

43 

18 

.45153 

2.21475 

.47270 

2.11.552 

.49423 

2.02:335 

.51614 

1.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11.392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47^41 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.495.32 

2.01891 

.51724 

1.9:3.332 

39 

22 

.45292 

2.20790 

'    .47412 

2.10916 

.49.568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20019 

i    .47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10000 

.49640 

2.01449 

.51&35 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49077 

2.01:302 

.51872 

1.92782 

35 

26 

.45-132 

2.20108 

.47555 

2.10284 

i   .49713 

2.01155  I 

.51909 

1.92645 

34 

27 

.45467 

2.199:38 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19769 

.47026 

2.09969 

.49786 

2.00862 

.51983 

1.92:371 

32 

29 

.4.5538 

2.19.599 

.47002 

2.09811 

.49822 

2.00715 

..52020 

1.92235 

31 

30 

.45573 

2.1^430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.477a3 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

32 

.45643 

2.19093 

.47769 

2.09341 

.49931 

2.00277 

.521:31 

.1.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52108 

1.91690 

27 

3i 

.45713 

2.18755 

.47840 

2.09028- 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47913 

2.08716 

.5007-6 

1.99695 

.52279 

1.91282 

24 

37 

.45819 

2.18251 

.47948 

.2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

1.99406 

.52:353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17.582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07'630 

.50331 

1.98684 

..525.38 

1.903:37 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

'    .50.368 

1.98.540 

. 52575 

1.90203 

16 

45 

.46101 

2.16917  ' 

.482:34 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751  1 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.899:35 

14 

47 

.46171 

2.16585  ' 

.48.306 

2.07014 

.50477 

1.98110 

.52687 

1.8G801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.162.55 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89.533 

11 

50 

.46277 

2.16090  ! 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.8926b 

9 

52 

.46:348 

2.15760 

.48486 

2.06247 

.50660 

1.97:395. 

.52873 

1.89133 

81 

53 

.46:383 

2.15596 

.48521 

2.06094 

1   .50696 

1.97253 

.52910 

1. 89000 V 

7' 

54 

.4^118 

2.154:32 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.887.34 

5I 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

..50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05:3:33 

.50879 

1.90544 

.53096 

1.88337 

2 

59 

.46595 

2.14614  1 

.487:37 

2.05182 

.50916 

1.96402 

.53ia4 

1.88205 

1 

60 

„ 

.46631 
Cotaii^ 

2.144.51   j 
Taug     1 

.48773 

2.0.50.30 

.509.53 
Cotang 

1.96201 

.53171 
Cotang 

1.88073 

0 
/ 

Cotang 

Taug 

Tang    } 

Tang 

65°           1 

1            64°           ' 

i           6 

3°           1 

6 

2» 

TABLE  XII.-TANGENTS  AND  COTANGENTS. 


327 


28° 

1          .  29° 

30°            ' 

31° 

60 

Tang 
.53171 

Cotang 

Tang 
.55431 

Cotang 

Tang 
.57735 

Cotang  ' 
1.73205  1 

Tang 
.60086 

Cotang 
1.66428 

1.88073  ' 

1.80405  i 

1 

.53208 

1.87941 

!   .55469 

1.80281 

.57774 

1.73089  ' 

.60126 

1.66.318 

59 

2 

.53246 

1.8';  809  ; 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.60209 

58 

3 

.53283 

1.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1 . 06099 

57 

4 

.53320 

1.87546  1 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.0.5990 

56 

5 

.53358 

1.87415  i 

.55621 

1.79788 

,   .57929 

1.72625  i 

.60284 

1.65881 

55 

6 

.5:3395 

1.87283  1 

.55659 

1.79665 

!   .57968 

1.72509  1 

.60324 

1.65772 

54 

7 

.53432 

1.87152  t 

.55697 

1.79542 

]    .58007 

1.72393 

.60304 

1.65668 

53 

8 

.53470 

1.87021   ' 

,55736 

1.79419 

I   .58046 

1.72278 

.60403 

1.6.5554 

.52 

9 

.53507 

1.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55813 

1.79174 

:    .58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71933 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

1   .58201 

1.71817  ! 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55904 

1.7B685 

.58279 

1.71588 

.00642 

1.64903 

46 

15 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

,    .58:357 

1.71358 

.60721 

1.04687 

44 

17 

.53807 

1.85850 

.56079 

1.78:319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

1.85720  i 

.56117 

1.78198 

,   .58435 

1.71129 

.60801 

1.04471 

42 

19 

.53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.04256 

40 

21 

.53957 

1.853.33 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.04148 

39 

22 

.53995 

1.85204  '■ 

.56270 

1.77713 

..58591 

1.70673 

.60960 

1.04041 

38 

23 

.54032 

1.85075 

.56309 

1.77592 

.58631 

1.70560 

.61000 

1.0.39:34 

37 

24 

.54070 

1.84946 

.56:^47 

1.77471 

.58670 

1.70416 

.61040 

1.63826 

36 

25 

.54107 

1.84818 

.56335 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

1.84689 

.56424 

1.77230 

.58748 

1.70219 

.61120 

1.6.3612 

34 

27 

.54183 

1.84561   ! 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

1.84433 

.56501 

1.76990 

.58826 

1.69992 

.61200 

1.63.398 

32 

29 

.54258 

1.84305 

.56539 

1.76869 

i   .58865 

1.69879 

.61240 

1.6.3292 

31 

30 

.54296 

1,84177 

.56577 

1.76749 

1   .58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

i    .56616 

1.76629 

.58944 

1.69653' 

!    .61320 

1.63079 

29 

32 

.54371 

1.83922 

.56654 

1.76510 

1   .58983 

1.09541 

1    .61360 

1.62972 

28 

33 

.54409 

1.8;i794 

.56693 

1.76390 

'   .59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

.59001 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.  O'iO/w'V 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83280 

.50846 

1.75913 

.59179 

1.68979 

.61501 

1.62442 

23 

38 

.54597 

1.83159 

.50885 

1.75794 

.59218 

1.68866 

..61601 

1.62336 

22 

39 

.54635 

1.83033  j 

.56923 

1.75675 

'    .59258 

1.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.68043 

.61681 

1.02125 

20 

41 

.54711 

1.82780 

.57000 

1.75437 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.570:30 

1.75:319 

.59:376 

1.68419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1  61808 

17 

44 

.54824 

1.82402 

.57116 

1.75082 

.59454 

1.68190 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

:    .59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

i   .59533 

1.67974 

.61922 

1.61493 

14 

47 

.54938 

1.82025 

.572:32 

1.74728 

1   .59573 

1.67863 

61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.07752 

.62003 

1.61283 

12 

49 

.5.5013 

1.81774 

'   .57:309 

1.74402 

.59651 

1. 67641 

.62043 

1.61179 

11 

50 

.55051 

1.81649 

.57:348 

1.7'4375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

; 55089 

1.81524 

.57:386 

1.74257 

'    .59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67:309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

i   .59809 

1.67198 

.62204 

1.60761 

t 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.7:3788 

.59888 

1.66978 

.622a5 

1.00553 

5 

56 

.55279 

1.80901  : 

.57-580 

1.7:3671 

.59928 

1.66867 

.62:325 

1.60449 

4 

57 

.55317 

1.S0777  1 

.57619 

1.73555 

'   .59967 

1.66757 

.62366 

1.60.345 

3 

58 

.  55:i55 

1.S0653  I 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

50 

.55393 

1.80529 

'   .57696 

1.7:3:321 

.60046 

1.66538 

.62446 

1.60137 

1 

60 

.554:J1 
Cotaijg 

!           6 

1.M0405  i 
Tang     ; 

i° 

1   .57735 
Cotant^ 

G 

1.73205 
Tang 

I 

0^              ! 

.60086 
Cotang 

1.66428 

.62487 
Cotang 

1.600:33 

0 

Tang 

Tang 

6 

9° 

5 

8° 

rA.TA"A* 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


/ 
~0 

32°            i 

33°            11 

34°             1 

35°           1 

/ 
60 

Tang  1 
.6^487 

Cotangr 
1.60033 

Tang 
.64941 

Cotang 

Tang 

Cotang   i 

Tang 
.70021 

Cotang 

1.53986 

.67451 

1.48256 

1.42815 

1 

.62527 

1.. 59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

2 

.62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

5 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42,374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70388 

1.42110 

52 

9 

.628.52 

1.59105 

.6.5314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

48 

13 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

.63095 

1.58490 

.65563 

1.52525 

.08088 

1.46870 

.70673 

1.41497 

45 

16 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

ir 

.63177 

1.58286 

.65646 

1.52332  ' 

.68173 

1.46686 

.70760 

1.41.322 

43 

18 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51940 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51850 

.68386 

1.46229 

.70979 

1.40887 

.38 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

37 

24 

.63462 

1.5(Oio 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.65980 

1.51562 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

.63544 

1.57372 

.66021 

1.51466 

.68557 

1.45864 

.71154 

1.40540 

34 

27 

.63584 

1.572^1 

.66063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45682 

.71242 

1.40.367 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.68685 

'1.4.5592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.4.5410 

.71373 

1.40109 

29 

32 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.4.5229 

.71461 

1.39936 

27 

M 

.63871 

1.56566 

.66356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

26 

35 

.63912 

1.56460 

.66398 

1.50607 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.. 56366 

.66440 

1.50512 

.08985 

1.44958 

.71593 

1.. 39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417  1 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322  1 

.09071 

■  1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.560G5 

.66566 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.. 39250 

19 

42 

.64199 

1.55766 

.66092 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.. 38909 

15 

46 

.64;3G3 

1.55308 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.4.3881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

.60986 

1.49284 

.69545 

1.43792 

.72167 

1. 38.508 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.384&4 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.4.3525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56 

.64775 

1.54379 

I   .67282 

1.48629 

.69847 

1.4.3169 

.7^77 

1.37976 

4 

57 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

5S 

.64858 

1.54183 

.67366 

1.48442 

.09934 

1.42992 

.72505 

1.37807 

2 

5S 

.64899 

1.54085 

.67409 

1.48.349 

.69977 

1 .42903 

.72610 

1.37722 

1 

6C 

1     .64941 
Cotang 

1.. 53986 

i   .67451 

1.48256 

.70021 

1.42815 

.72654 
Cotang 

1.376.38 

0 

> 
J 

1    Tang 

Cotang 

Tang 

Cotang 

Tang 

Tang 

57° 

i            56° 

55° 

54° 

TABLE  XII. -TANGENTS  AND  COTANGENTS. 


329 


"o 

36°           1 

37°            1 

38° 

39°           1 

/ 

60 

Tang  1 
.72654 

■ 
Cotang 

Tang  i 
.75355 

Cotang 
1.32704 

Tang  1 
.78129 

Cotang 

Tang  1 
.80978 

Cotang 
1.2:3490 

1.37638 

1.27994 

1 

.72699 

1.37554 

.75401 

1.32024 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32404 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

72832 

1.37302 

.75538 

1.32384 

.78:316 

1.27688 

.81171 

1.23196 

66 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36907 

.75721 

1.32004 

.7.S504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.3()883 

.75707 

1.31984 

.78551 

1.27306 

.81413 

1.228:51 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.225:59 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.30S83 

.76042 

1.31507 

.78834 

1.20849 

.81703 

1.22394 

45 

16 

.73308 

1.36300 

.76088 

1.31427 

.78881 

1.20774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.761M 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73502 

1.360.51 

.76226 

1.31190 

.79022 

1.26546  ! 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.219.59 

39 

32 

.73637 

1.35802 

.76364 

1.30952 

.79104 

1.26319  ' 

.82044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244  i 

.82092 

1.21814 

37 

21 

.73726 

1.35037 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.tOIll 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73801 

1.35389  i 

.76594 

1.305.58 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.70040 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.70086 

1.30401 

.79496 

1.25702 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

i. 30323 

.79544 

1.25717 

.824:34 

1.21310 

30 

51 

.74041 

1.3.5060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.70825 

1.30166 

.79639 

1.25567 

.82531 

1.21100 

28 

3:5 

.74131 

1.34896 

.70871 

1.30087 

.79086 

1.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.797:34 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

l.;W732 

.70964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268  1 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193  i 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20605 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.204.51 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74074 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

4G 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.8:5218 

1.20106 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024 

12 

49 

.74855 

1.3.3.592 

.77615 

1.28842 

.804.50 

1.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80.546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

75037 

1.33268 

.77'801 

1.28533 

.80042 

1.24005 

.83564 

1.19069 

7 

54 

.75082 

1.3;5187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28:579 

.807:38 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.3^3026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.808:34 

1.2:3710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.2:5563 

.83860 

1.19iM6 

1 

00 

.75:555 
jCotang 

1.32704 

.78129 
Cotang 

1.27994 
Tang 

.80978 
Cotang 

1.2:3490 
Tang 

.83910 
Cotang 

1.19175 

_0 

Tang 

Tang 

63° 

52° 

51° 

1           60° 

330 


TABLE  XIl.— TANGENTS  AND   COTANGENTS. 


40° 

41° 

42° 

43°            1 

60 

Tang     Cotang 

Tang 
.86929 

Cotang 
1.15037 

Jang 
.90040 

Cotang 
1.11061 

Tang 
.93252 

Cotang 

.83910 

1.19175 

1.07237  1 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

2 

.8W09 

1.1903.5 

.87031 

1.14902 

.90146 

1.109:31 

^  .93360 

1.07112  1 

58 

3 

.^059 

1.18964 

1   .87082 

1.148*4 

.90199 

1.10^67  i 

.93415 

1.07049 

57 

4 

.81108 

1.18894 

1    .87ia3 

1.14767 

.90251 

1.10802  1 

.9*469 

1.06987 

56 

5 

.84158 

1.18824 

'   .87184 

1.14699 

.90:304 

1.107:37 

.9:3524  ■ 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14632 

.90:357 

1.10672 

.93578 

1.06862 

54 

7 

.84:',t8 

1.18684 

.87287 

1 .  14565 

.90410 

1 . 10607 

.93633 

1.06800 

53 

8 

.8*307  i 

1.18614 

.87338 

1.14498 

.90463 

1.10543 

.93688 

1.06738 

52 

9 

.84;357  1 

1.1S544 

i    .87389 

1.144:30 

.90516 

1.10478 

.93742 

1.06676 

51 

10 

.81407 

1.18474 

:    .87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457  j 

1.18404  i 

.87492  , 

1.14296 

.90621 

1.10349 

.93852 

1.06551 

49 

12 

.84507  1 

1.18334 

.87543 

1.14229 

.90674 

1.10285 

.93906 

1.06489 

■fe 

13 

.84556 

1 . 182G4 

.87595  i 

1.14162 

.90727 

1.10220 

.93961 

1.06427    47 

14 

.&4606 

1.18194 

.87646 

1.14095 

.90781 

1.10156 

.94016 

1.06365    46 

15 1 

.84656 

1.18125 

.87698 

1.14028 

.90834 

1 . 10091 

.94071 

1.06303    45 

16: 

.84706  ; 

1.18055 

.87749      1.13961 

.90887 

1.10027 

.94125 

1.06241    44 

17 

.84756  i 

1.17986 

.87801      1.1:3894 

.90940 

1.09963 

.94180 

1.06179    43 

18 

.84806 

1.17916 

.87852      1.13828 

.90993 

1.09899 

.94235 

1.06117 

42 

19 

.84856 

1.17^46 

.87904 

1.13761 

.91046 

1.09^34 

.94290 

1.06056 

41 

20 

.84906 

1 . 17777 

. 87955 

1.13694 

.91099 

1.09770 

.94345 

1.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006      1.17638 

.88059 

1.1:3561 

.91206 

1.09(>42 

.94455 

1.05870 

38 

23 

.85057      1.17569 

.88110 

1.1:3494 

.91259 

1.09578 

.94510 

1.05809    37 

^ 

.85107 

1.17500 

.88162 

1.1:3428 

.91:313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.174:30 

.88214      1.13:361 

.91366 

1.09450 

.94620 

1.05685 

35 

26 

.85207 

1.17361 

.88265      1.1:3295  ' 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.a5-:'o7 

1.17292 

.88^317     1.1:3228 

.91473 

1.09.322 

.94731 

1.05562 

33 

28 

.85308 

1.17223 

.aS369      1.131G2 

.91526 

1.09258 

.94786 

1.05501 

32 

29 

.85358 

1.171.54 

.88421      1.1:3096 

'    .91580 

1.09195 

.94&41 

1.054.39 

31 

30 

.85408 

1.17085 

!  .8t^73 

1.13029 

.9163:3 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.05317 

29 

32. 

.85509 

1.1G947 

.88576 

1.12897 

!    .91740 

1.09003 

.95007 

1.052.55 

28 

33 

.85559 

1.16878 

.88628 

1.128:31 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1.16809 

;   .88680 

1.12765 

.91847 

1.08876 

.95118 

1.05133 

26 

35 

.85660 

1.16741 

;   .88732 

1.12699 

.91901 

1.08813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

1   .88784 

1 .  126:33 

91955 

1.08749 

.95229 

1.05010 

24 

37 

.85761 

i.ieeas 

.888:36 

1.12567 

.92008 

1.08686 

.95284 

1.04949 

23 

38 

.85811 

1.1G535 

.88888 

1.12501 

.92062 

1.08622 

.95.^40 

1.04888 

22 

39 

.85862 

1.1G4G6 

i   .88940 

1.124:35 

.92116 

1.08559 

.95.395 

1.04827 

21 

40 

.85912 

1.16398 

1   .88992 

1.12369 

.92170 

1.08496 

.95451 

1.04766 

20 

41 

.85963 

1.16.329 

.89045 

1.12.303 

.92224 

1.084.32 

.95506 

1.04705 

19 

42 

.86014 

1.1G261 

!    .89097 

1.122:3s 

.92277 

1.08:369 

.95562 

1.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92:331 

1.08306 

.95618 

1.04.5a3 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92:385 

1.08243 

.95673 

1.04.522 

16 

45 

.86166 

1.16056 

i   .892.53 

1.12041 

92439 

1.08179 

.95729 

1.04461 

15 

46 

.86216 

1.159S7 

.89:306 

1.11975 

.92493 

1.08116 

.95785 

1.04401 

14 

47 

.86207 

1.15919 

.89:358 

1.11909 

.92^7 

1.08053 

.95841 

1.04.340 

13 

48 

.86318 

1.1.5851 

;   .89410 

1.11.844 

.92601 

1 .07990 

.95897 

1.04279 

12 

49 

.86.368 

1.15783 

1    .89463 

1  11778 

.92655 

1.07927 

.95952 

1.04218 

11 

50 

.86419 

1.15715 

;    .89515 

1.11713 

.92709 

1.07864 

;   .96008 

1.04158 

10 

51 

'    .8&470 

1.15647 

'   .89567 

1.11648 

'   .92763 

1.07801 

.96064 

1.04097 

9 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

1.040:36 

8 

53 

,   .86572 

1.15511 

'    .89672 

1.11517 

.92872 

1.07676 

.96176 

1.03976 

7 

54 

1   .86623 

'  1.1544:3 

:    .89725 

1.11452 

.92926 

1.07613 

.96232 

l.a3915 

6 

55 

'.86G74 

1 . 15375 

:|   .89777 

1.11387 

.92980 

1.07550 

.96288 

1.0:3855 

5 

56 

.86725 

i  1.15:308 

.89830 

1.11.321 

.9:30:34 

1.07487 

.96*44 

1.03794 

4 

57 

.86776 

1.15240 

:     .89883 

1.112.56 

.9308.8 

1.07425 

.96400 

1.0.37:34 

3 

58 

.86827 

1.15172 

!|   .899.35 

1.11191 

.9:3143 

1.07.362 

1   .964.57 

1.0:3674 

2 

59 

.86878 

1.1.5104 

.89988 

1.11126 

.9:3197 

1.07299 

'   .96513 

1.0:3613 

1 

60 

1 

.86929 
Cotang 

1  1.tO:37 
Tang 

.90040 

1.11061 

.93252 
Cotang 

4 

1.07237 
Tang 

70 

1   .96.569 
Cotang 

1  03553 

0 

'  Cotang     Tang 

Tang 

49° 

4 

8° 

46° 

TABLE  XII. -TANGENTS  AND  COTANGENTS. 


331 


44° 

440 

1 

44° 

1    / 

t 

60 

/ 

1 

ao 

/ 

/ 

/ 

0 

Tang 

Cotang 
1.0:3553 

Tang  1  Cotang 

Tang 
.98843 

Cotang 

.96569 

.97700 

1.02:355 

40 

40 

1.01170 

20 

1 

.96625 

1.0:i493 

59 

21 

.97756 

1.02295 

39     41 

.98901 

1.01112 

19 

2 

.96681 

1.0:34:3:3 

58  i 

22 

.97813 

1.02236 

.38     42 

.98958 

1.01053 

18 

3 

.967:58 

1.0:3:372 

57  1 

23 

.97870 

1.02176 

37     4:3 

.99016 

1.00994 

17 

4 

.9679^1 

1.0:3:312 

56  1 

24 

.97927 

1.02117 

:36     44 

.99073 

1.009:35 

16 

5 

.968.50 

1.0:3252 

00 

25 

.97984 

1.02057 

:3.5     45 

.99131 

l.(X).S76 

15 

6 

.96907 

1.0:3192 

54  1 

26 

.98041 

1.01998 

34     46 

.99189 

l."0818 

14 

7 

.96963 

1.031:32 

53  j 

27 

.98098 

1.019:39 

:3:3     47 

.99247 

1.00759 

13 

8 

.97020 

1.0:3072 

52 

28 

.981.55 

1.01879 

:32     48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

31     49 

.99362 

1.00642 

11 

10 

.9713:3 

1.02952 

50 

30 

.98270 

1.01761 

30  |50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98:327 

1.01702 

29     51 

.99478 

1.00.525 

9 

12 

.97246 

1.028^ 

48 

32 

.98384 

1.01642 

28     52 

.99536 

1.00467 

8 

13 

.97:302 

1.02772 

47 

3:3 

.98441 

1.01.583 

27     53 

.99594 

1.00408 

1^ 
i 

14 

.97:359 

1.02713 

46 

.34 

.98499 

1.01.524 

26     54 

.99652 

1.0a3.5<T 

6 

15 

.97416 

1.02653 

45  1 

35 

.98556 

1.01465 

25     55 

.99710 

1.00291 

5 

16 

.97472 

1.02593 

44  1 

36 

.98613 

1.01406 

24     56 

.99768 

1.002.33 

4 

17  1 

.97529 

1.02.5.33 

43  ' 

37 

.98671 

1.01347 

2:3     57 

.99826 

1.00175 

3 

18  1 

.97586 

1.02474 

42  i 

38 

.98728 

1.012S8 

22     58" 

.99884 

1.00116 

2 

19 

.97643  1 

1.02414 

41  i 

39 

.98786 

1.01229 

21     59 

.99942  ' 

1.00058 

1 

20    ; 

.97700  ! 

1.02:355 

40 

40 

/ 

.98843 

1.01170 

20 

60 

1.00000  1 

1.00000 

0 

/  { 

Cotang 

Tang 

/ 

Cotang  1 

Tang    j 

/ 

/ 

Cotang  ; 

Tang 

/ 

45° 

45° 

46° 

s/ersine::  {-oa6\  E^ssecarrt^Secc^nf  ") 


332 

TABLE 

xin.-^ 

rERSIN] 

ES  AKD 

EXSEC 

ANTS. 

1 

0 

s 

1 

0 

2 

i 
0 

3 

0 

f 
0 

Vers, 

Exsec. 

Vers. 

Exsec.  1 

1 

Vers. 

i 
Exsec.  ' 

Vers. 

Exsec, 

0 

.00000 

.00000 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

1 

.00000 

.00000 

.00016 

.00016 

.00062 

.00062 

.001:39 

001:39 

1 

2 

.00000 

.00000 

.00016 

.00016  i 

.00063 

.00063 

.00140 

.00140 

2 

3 

.00000 

.00000  ; 

.00017 

.00017  1 

.00064 

.00064 

.00142 

.00142 

3 

4 

.00000 

.00000  ; 

.00017 

.00017  i 

.00065 

.00065 

.0014:3 

.00143 

4 

5 

.00000 

.00000  ' 

.00018 

.00018 

.00066 

.00066 

.00145 

.00145 

5 

6 

.00000 

.(moo 

.00018 

.00018 

.00067 

.00067 

.00146 

.00147 

6 

7 

.00000 

.000<X)  ' 

.00019 

.00019 

.00068 

.00068 

.00148 

.00148 

7 

8 

.00000 

.00000 

.00020 

.00020 

.00069 

.00069 

.00150 

.00150 

8 

9 

.aiooo 

.00000 

.00020 

.00020 

.00070 

.00070 

.00151 

.00151 

9 

10 

.00000 

.00000 

.00021 

.00021 

.00071 

.90072 

.00153 

.00153 

10 

11 

.00001 

.00001 

.00021 

.00021 

.00073 

.00073 

!00154 

.00155 

11 

12 

.00001 

.00001 

.00022 

.00022 

.00074 

.00074 

.00156 

.00156 

12 

13 

.00001 

.00001 

.00023 

.00023 

.00075  1 

.00075 

.001.58 

.00158 

13 

14  ! 

.00001 

.00001 

.00023 

.00023 

.00076 

.00076 

.00159 

.00159 

14 

15 

.00001 

.00001 

.00024 

.00024 

.00077 

.00077 

.00161 

.00161 

15 

16 

.00001 

.00001 

.00024 

.00024 

.00078 

.00078 

.00162 

.00163 

16 

IT ; 

.00001 

.00001 

.00025 

.00025 

.00079 

.00079 

.00164 

.001&4 

17 

18  1 

.00001 

.00001  1 

.00026 

.00026 

.00081 

.00081 

.00166 

.00166 

18 

19  ' 

.00002 

.00002 

.00026 

.00026 

.00082 

.00082 

.00168 

.00168 

19 

20 

.00002 

.00003 

.00027 

.00027 

.00083 

.00083  i 

.00169 

.00169 

20 

21 

.00002  * 

.00002 

.00028 

.00028 

1  .00084 

.00084 

.00171 

.00171 

21 

22 

.00002 

.00002 

.00028 

.00028 

:  .00085 

.00085 

.00173 

.00173 

22 

23 

.00002 

.00002 

.00029 

.00029 

;  .00087 

.00087 

.00174 

.00175 

23 

24 

.00003 

.00002 

.00030 

.00030 

1  .00088 

.00088 

.00176 

.00176 

24 

25 

.00003 

.00003 

.00031 

.00031 

.00089 

.00089 

.00178 

.00178 

25 

26 

.00003 

.00003  ' 

.00031 

.00031 

!  .00090 

.00090 

.00179 

.00180 

26 

27 

.00003 

.00003 

.00032 

.00032 

1  .00091 

.00091 

.00181 

.00182 

27 

28 

.00003 

.00003 

.000:3:3 

.0003:3  1 

.00093 

.00093  ; 

.001  as 

.00183 

28 

29 

.00004 

.00004 

.00034 

.00034 

i  .00094 

.00094  ' 

.001^5 

.00185 

29 

30 

.00004 

.00004 

.00034 

.00034 

.00095 

.00095 

.00187 

.00187 

30 

31 

.00004 

.00004 

.00035 

.00035 

.00096 

.00097  1 

.00188 

.00189 

31 

32 

.0(XW4 

.00004 

.00036 

.00036 

.00098 

.00098  ' 

.00190 

.00190 

32 

3:3 

.00005 

.00005 

.000:37 

.000:37 

.00099 

.00099 

.00192 

.00192 

3:3 

34 

.00005 

.00005 

.00037 

.00037 

.00100 

.00100 

.00194 

.00194 

34 

35 

.00005 

.00005 

.000:38 

.000:38 

.00102 

.00102 

.00198 

.00196 

35 

36 

.00005 

.ooa)5 

.00039 

.000:39 

.00103 

.00103 

.00197 

.00198 

36 

37 

.00006 

.00006 

.00040 

.00040 

.00104 

.00104 

.00199 

.00200 

37 

38 

.00006 

.00006  i 

.00041 

.00041 

.00106 

.00106 

.00201 

.00201 

38 

39 

.00006 

.ax)06  ■ 

.00041 

.00041  1 

;  .00107 

.00107 

.00203 

.00203 

39 

40 

.00007 

.00007 

.00042 

.00042  ■ 

.00108 

.00108 

.00205 

.00205 

40 

41 

.00007 

.00007 

.00043 

.00043 

.00110 

.00110 

.00207 

.00207 

41 

42 

.00007 

.00007 

.00044 

.00044 

.00111 

.00111 

.00208 

.00209 

42 

43 

.00008 

.00008 

.00045 

.00045 

.00112 

.00113 

.00210 

.00211 

43 

44 

.00008 

.00008 

.00046 

.00046 

.00114 

.00114 

1  .00212 

.00213 

44 

45 

.00009 

.00009 

.00047 

.00047 

.00115 

.00115 

.00214 

.00215 

45 

46 

.00009 

.00009 

.aX)48 

.00048 

i  .00117 

.00117 

.00216 

.00216 

46 

47 

.00009 

.00009 

.00048 

.00048 

.00118 

.00118 

.00218 

.00218 

47 

48 

.00010 

.00010 

.00049 

.00049 

.00119 

.00120 

.00220 

1  .00220 

48 

49 

.00010 

.00010 

.00050 

.00050 

.00121 

.00121 

.00222 

.00222 

49 

50 

.00011 

.00011 

.00051 

.00051 

.00122 

.00122 

.00224 

.00224 

50 

51 

.00011 

.00011 

.00052 

.00052 

.00124 

.00124 

.00226 

.00226 

51 

52 

.00011 

.00011 

.000.53 

.000.53 

.00125 

.00125 

.00228 

.00228 

52 

5;} 

.00012 

.00012 

.00054 

.000,>4 

.00127 

;  .00127 

.00230 

.002:30 

53 

54 

.00012 

.00012 

.000.55 

.00055 

;  .00128 

.00128 

.002:32 

.002:32 

54 

55 

.00013 

.00013 

.00056 

.000.56 

.001:30 

.001:30 

.002:34 

.002:34 

55 

56 

.00013 

.00013 

.00057 

.000.57 

.00131 

i  .001:31 

.002:36 

.002:36 

56 

57 

.00014 

.00014 

.000.58 

.000.58 

.00133 

I  .001:3:3 

.00238 

.00238 

57 

58 

.00014 

.00014 

.00059 

.00059 

.001:34 

.001:34 

.00240 

.00240 

58 

59 

.00015 

.00015 

.  .0<¥)60 

.00060 

.00136 

.001:36 

.00242 

.00242 

59 

60 

.00015 

.00015 

1  .00061 

.00061 

.00137 

.00137 

:  .00244 

.00244 

60  1 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


333 


~o" 

40 

»° 

6° 

7 

0 

0 

Vers. 

Exsec. 

1 

Vers. 
.00381 

Exsec. 

.00382 

Vers. 

1 
Exsec. 

Vers. 

Exsec. 

.00244 

.00244  ' 

.00548 

.00551  1 

.00745 

.00751 

1 

.00246 

.00246  i 

.00383 

.00385  1 

.00551 

.00554  i 

.00749 

.00755 

1 

o 

.00248 

.00248 

.00386 

.00387  ! 

.00554 

.00557 

.00752 

.00758 

2 

3 

.00250 

.00250 

.00388 

.00390 

.00557 

.00560  1 

.00756 

.00762 

3 

4 

.00252 

.00252  ' 

.00391 

.00392 

.00560 

.00563 

.00760 

.00765 

4 

5 

.00254 

.00254 

.00393 

.00395  i 

.00563 

.00566 

.00763 

.00769 

5 

6 

.00256 

.00257 

.00396 

.00397  1 

.00566 

.00569 

.00767 

.00773 

6 

7 

.00258 

.00259 

.00398 

.00400  i 

.00569 

.00573 

.00770 

.00776 

7 

8 

.002(50 

.00261  \ 

.00401 

.00403  i 

.00572 

.00576  ■ 

.00774 

.00780 

8 

9 

.00202 

.002(53 

.00404 

.00405  1 

.00576 

.00579  1 

.00778 

.00784 

9 

10 

.00264 

.00265 

.00406 

.00408 

.00579 

.00582  ! 

.00781 

.00787 

10 

11 

.00206 

.00207 

.00409 

.00411  1 

00582 

.00585  ! 

.00785 

.00791 

11 

12 

.00269 

.002(59  1 

.00412 

.00413 

.00585 

.00588  i 

.00789 

.00795 

12 

13 

.00271 

.00271 

.00414 

.00416  ; 

.00588 

.00592  ! 

.00792 

.00799 

13 

14 

.00273 

.00274 

.00417 

.00419 

.00591 

.00595  i 

00796 

.00802 

14 

15 

.00275 

.00276 

.00420 

.00421  ! 

00594 

..00598 

.00800 

.00806 

15 

16 

.00277 

.00278 

.00422 

.00424  1 

.00598 

.00601 

.00803 

.00810 

16 

17 

.00279 

.00280 

.00425 

.00427  ' 

.00601 

.00604 

.00807 

.00813 

17 

18 

.00281 

.00282 

.00428 

.00429  ; 

00604 

.00608  ; 

.00811 

.00817 

18 

19 

.00284 

.00284 

.00430 

.00432 

.00607 

.00611 

.00814 

.00821 

19 

20 

.00286 

.00287 

.00433 

.00435 

.00610 

.00614 

.00818 

.00825 

20 

21 

.00288 

.00289 

.0043(5 

.00438  i 

.00614 

.00617 

.00822 

.00828 

21 

22 

.00290 

.00291  ' 

.(X)438 

.00440  ; 

.00617 

.00621  1 

.00825 

.00832 

23 

23 

.00293 

.00293 

.00441 

.00443  i 

.00620 

.00624 

.00829 

.00836 

23 

24 

.00295 

.00296 

.00444 

.00446 

.00623 

.00627  ! 

.00833 

.00840 

24 

25 

.00297 

.00298 

.00447 

.00449 

.0062(5 

.00630  ; 

.00837 

.00844 

25 

2(5 

.00299 

.00300  1 

.00449 

.00451  ! 

.00630 

.00634  ! 

.00840 

.00848 

26 

27 

.00301 

.00302  1 

.00452 

.00454  i 

.00633 

.00637 

.00844 

.00851 

27 

28 

.00304 

.00305 

.00455 

.00457 

.00636 

.00640 

.00848 

.00855 

28 

29 

.00306 

.00307  ; 

.00458 

.00460 

.00640 

.00644 

.00852 

.00859 

29 

30 

.00308 

.00309 

.00460 

.00463  , 

.00643 

.00647 

.00856 

.00863 

30 

31 

.00311 

.00312  , 

.00463 

.00465  , 

.00646 

.00650 

.00859 

.00867 

31 

32 

.00313 

.00314  I 

.00466 

.00408 

.00649 

.00654 

.00863 

.00871 

32 

33 

.00315 

.00316  i 

.00469 

.00471 

.00653 

.00657 

.00867 

.00875 

33 

34 

.00317 

.00318 

.00472 

.00474 

.00656 

.00660  \ 

.00871 

.00878 

34 

35 

.00320 

.00321 

.00474 

.00477 

.00659 

.00664  : 

.00875 

.00882 

35 

36 

.00323 

.00323  i 

.00477 

.00480  1 

.006(53 

.00667  ; 

.00878 

.00886 

36 

37 

.00324 

.00326  ! 

.00480 

.00482 

.00666 

,00671 

.0(J882 

.00890 

37 

38 

.00327 

.00328  i 

.00483 

.00485  1 

.00669 

.00674  ! 

.00886 

.00894 

38 

39 

.00329 

.00330  i 

.00480 

.00488  ; 

.00673 

.00677  I 

.00890 

.00898 

39 

40 

.00333 

.00333  ! 

.00489 

.00491 

.00676 

.00681  1 

.00894 

.00902 

40 

41 

.00334 

.00335  j 

.00492 

.00494 

.00680 

.00684 

.00898 

.00906 

41 

42 

.00336 

.00337 

.00494 

.00197 

.00683 

.00(588 

.00902 

.00910 

42 

43 

.00339 

.00340 

.00497 

.00500 

.00686 

.00691 

.00906 

.00914 

43 

44 

.00341 

.00342  ' 

.00500 

.00503  i 

00690 

.00695 

.00909 

.00918 

44 

45 

.00343 

.00345  ' 

.00503 

.00506  ; 

.00693 

.00698 

00913 

.00922 

45 

46 

.00346 

.00347 

.00506 

.00509  ■ 

.00697 

.00701 

00917 

.00926 

46 

47 

.00348 

.00350  1 

.00509 

.00512  1 

.00700 

.00705 

.00921 

.00930 

47 

48 

.00351 

.00352  ' 

.00512 

.00515  ; 

.00703 

.00708 

.00925 

.00934 

48 

49 

.00353 

.00354 

.00515 

.00518 

.00707 

.00712 

00929 

.00938 

49 

50 

.00350 

.00357 

.00518 

.00521  1 

.00710 

.00715  ; 

.00933 

.00942 

50 

51 

.00358 

.00359 

.00521 

.00524  j 

.00714 

.00719 

.00937 

.00946 

51 

52 

.00361 

.00362 

.00524 

.00527 

.00717 

.00722 

.00941 

.00950 

52 

53 

.003(53 

.00364 

.00527 

.00530 

.00721 

.00726 

.00945 

.00954 

53 

54 

.00365 

.003(57 

.00530 

.00533 

.00724 

.00730 

.00949 

.00958 

54 

55 

.00368 

.00369 

.00533 

.00536 

.00728 

.007;ij 

.00953 

.00962 

55 

56 

.00370 

.0037'2 

.00.536 

.00539 

.00731 

.00737 

.00957 

.00966 

56 

57 

.00373 

.00374 

.00539 

.00.^)42 

.00735 

.00740 

.00961 

.00970 

57 

58 

.00375 

.00377 

.00542 

.00545  ! 

.00738 

.00744 

.00965 

.00975 

58 

59 

.0(J378 

.00379 

.00.545 

.00548  1 

.00742 

.00747 

.009(59 

.00979 

59 

60 

.00381 

.00382 

.00548 

.00551  1 

.00745 

.00751 

.00973 

.00983 

60 

'#ArA~Av  r 


TABLE   XIII.-VERSINES   AND  EXSECANTS. 


1 
0 

8 

o 

9 

o 

lo- 

11 

0 

~0 

Vers. 

Exsec. 

Vers. 

Exsec. 

vers. 

Exsec. 

Vers. 

Exsec. 

.00973 

.00983  : 

.01231 

.01247 

.01.519 

.01543 

.01837 

.01872 

1 

.00977 

.00987 

.01236 

.01251 

1  .01.524 

.01548  1 

.01843  j 

.01877 

1 

2 

.00981 

.00991  : 

.01240 

.01256 

\   .01529 

.01553 

.01848 

.01883 

2 

3 

.00985 

.00995  ! 

.01245 

.01261 

.01534 

.01558 

.01854 

.01889 

3 

4 

.00989 

.00999  , 

.01249 

.01265 

.01540 

.01564 

.01860 

.01895 

4 

5 

.00994 

.01004 

.01254 

.01270 

.01545 

.01569 

.01865 

.01901 

5 

6 

.00998 

.01008 

.01259 

.01275  I 

.01.550 

.01574 

.01871 

.01906 

6 

1 

.01002 

.01012 

.01263 

.01279 

.01555 

.01579 

.01876 

.01912 

7 

8 

.01006 

.01016 

.01268 

.01284 

.01.560 

.01585 

.01882 

.01918 

8 

9 

.01010 

.01020  ! 

.01272 

.01289 

.01.565 

.01590 

.01888 

.01924 

9 

10 

.01014 

.01024 

.01277 

.01294 

.01570 

.01595 

.01893 

.01930 

10 

11 

.01018 

.01029 

.01282 

.01298 

.01575 

.01601 

.01899 

.01936 

11 

12 

.01022 

.01033  1 

.01286 

.01303 

.01580 

.01606  ; 

.01904 

.01941 

12 

13 

.01027 

.01037  1 

.01291 

.01:308 

.01586 

.01611 

.01910 

.01947 

13 

14 

.01031 

.01041  ' 

.01296 

.01313 

.01591 

.01616 

.01916 

.01953 

14 

15 

.010:35 

.01046 

.01300 

.01:318 

.01596 

.01622 

.01921 

.01959 

15 

16 

.01039 

.01050 

.01:305 

.01323 

.01601 

.01627 

.01927 

.01965 

16 

ir 

.01043 

.01054 

.01310 

.01327 

.01606 

.01633 

.019.33 

.01971 

17 

18 

.01047 

.01059 

.01:314 

.01332 

.01612 

.01638 

.01939 

.01977 

18 

19 

.01052 

.0106:3 

.01319 

.01:337  i 

.01617 

.01643 

.01944 

.019a3 

19 

20 

.01056 

.01067 

.01324 

.01342  i 

.01622 

.01649 

.01950 

.01989 

20 

21 

.01060 

.01071  ' 

.01329 

.01346  i 

.01627 

.01654  ' 

.01956 

.01995 

21 

22 

.01064 

.01076 

.013:33 

.01351 

.01632 

.01659  ! 

.01961 

.02001 

22 

23 

.01069 

.01080 

.01338 

.01356  1 

.01638 

.01665  ; 

.01967 

.02007 

23 

21 

.01073 

.01084 

.01343 

.01:361  ' 

.01643 

.01670  ; 

.01973 

.02013 

24 

25 

.01077 

.01089  1 

.01348 

.01366 

.01648 

.01676  ! 

.01979 

.02019 

25 

26 

.01081 

.01093 

.01352 

.01371 

.01653 

.01681  , 

.01984 

.02025 

26 

27 

.oias6 

.01097 

.01357 

.01376 

.01659 

.01687  1 

.01990 

.02031 

27 

28 

.01090 

.01102 

.01362 

.01:381 

'  .01664 

.01692 

.01996 

.020:37 

28 

29 

.01094 

.01106 

.01367 

.01:386 

;  .01669 

.01698  j 

.02002 

.02043 

29 

30 

.01098 

.01111 

.01371 

.01391 

1  .01675 

.01703 

.02008 

.02049 

30 

31 

.01103 

.01115 

.01376 

.01395 

'  .01680 

.01709  ; 

.02013 

.02055 

31 

32 

.01107 

.01119 

.01:381 

.01400 

.01685 

.01714  i 

.02019 

.02061 

32 

:« 

.01111 

.01124 

.01.386 

.01405 

.01690 

.01720  ' 

.02025 

.02067 

33 

34 

.01116 

.01128 

.01:391 

.01410 

i  .01696 

.01725 

.02031 

.02073 

34 

35 

.01120 

.01133 

.01396 

.01415 

i  .01701 

.01731 

.02037 

.02079 

35 

36 

.01124 

.011:37 

.01400 

.01420 

!  .01706 

.01736 

.02042 

.02085 

36 

37 

.01129 

.01142 

.01405 

.01425 

.01712 

.01742 

.02048 

.02091 

37 

38 

.01133 

.01146 

.01410 

.014:30 

i  .01717 

.01747  , 

.02054 

.02097 

38 

39 

.01137 

.01151 

.01415 

.014:35 

1  .01723 

.017.53 

.02000 

.02103 

39 

40 

.01142 

.01155 

.01420 

.01440 

.01728 

.01758 

.02066 

.02110 

40 

41 

.01146 

.01160 

.01425 

.01445 

.01733 

.01764 

.02072 

.02116 

41 

42 

.01151 

.01164 

.01430 

.014.50 

.017:39 

.01769 

.02078 

.02122 

42 

43 

.01155 

.01169 

.01435 

.01455 

.01744 

.01775 

.02084 

.02128 

43 

44 

.01159 

.01173 

.01439 

.01461 

.01750 

.01781 

.02090 

.02134 

44 

45 

.01164 

.01178 

.01444 

.01466 

;  .017.55 

.01786 

.02095 

.02140 

45 

46 

.01168 

.01182 

.01449 

.01471 

.01760 

.01792  ' 

.02101 

.02146 

46 

47 

.01173 

.01187 

.01454 

■01476 

.01766 

.01798 

.02107 

.02153 

47 

48 

.01177 

.01191  i 

.01459 

.01481 

.01771 

.01803 

.02113 

.02159 

48 

49 

.01182 

.01196 

.01464 

.01486 

.01777 

.01809 

.02119 

.02165 

49 

50 

.01186 

.01200  1 

.01469 

.01491 

.01782 

.01815 

.02125 

.02171 

50 

51 

.01191 

.01205 

.01474 

.01496 

.01788 

.01820 

.02131 

.02178 

51 

52 

.01195 

.01209 

.01479 

.01.501 

.01703 

.01826 

.021:37 

.02184 

52 

53 

.01200 

.01214 

.01484 

.01506 

.01799 

.018:32 

.02143 

.02190 

53 

54 

.01204 

.01219 

;  .01489 

.01.512 

.01804 

.018.37 

.02149 

.02196 

54 

55 

.01209 

.01223 

.01494 

.01.517 

.01810 

.01843  i 

.02155 

.02203 

55 

56 

.01213 

.01228 

.01499 

.01.522 

.01815 

.01849 

.02161 

.02209 

56 

57 

.01218 

.0123:3 

.01504 

.01.527  ! 

.01821 

.01854 

.02167 

.02215 

57 

58 

.01222 

.012:37 

.01.509 

.01.5:32 

.01826 

.01860 

.02173 

.02221 

58 

59 

.01227 

.01242 

.01514 

.015:37 

.01^32 

.01866 

.02179 

.02228 

59 

60 

.01231 

.01247 

.01.519 

.01.543 

.01837 

.01872 

.02185 

.022:34 

60 

TAPLE   XIir.-VERSINES  AND   EXSECANTS. 


335 


1 

0 

12° 

13° 

14° 

16° 

/ 
0 

Vers. 

1 
Exsec.  , 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

.02185 

.022.34  1 

.02.563 

.02630 

.02970 

.03061 

.03407 

.03528 

1 

.02191 

.02240 

.02570 

.02637 

.02977 

.03069 

.03415 

.03536 

1 

2 

.02197 

.02247 

.02576 

.02644 

.02985 

.03076  : 

.03422 

.03544 

2 

3 

.02203 

.02253 

.02583 

.02651 

.02992 

.03084 

.03430 

.03,5.52 

3 

4 

.02210 

.02259 

.02.589 

.02658 

.02999 

.03091  { 

.034.38 

.03560 

4 

5 

.02216 

.02266 

.02596 

.02665 

.03006 

.03099 

.03445 

.03568 

5 

6 

.02222 

.02272 

.02602 

.02672 

.03013 

.03106 

.0;3453 

.03,576 

6 

7 

.02228 

.02-279 

.02609 

.02679 

.03020 

.03114 

.03460 

.03.584 

7 

8 

.02234 

.02285 

.02616 

.02686 

.03027 

.03121 

.03468 

.03592 

8 

9 

.02240 

.02291 

.02622 

.02693 

.03034 

.03129 

.03476 

.03601 

9 

10 

.02246 

.02298 

.02629 

.02700 

.03041 

.03137 

.03483 

.03609 

10 

11 

.022.52 

.02304 

.02635 

.02707 

.03048 

.03144 

.03491 

.03617 

11 

12 

.02258 

.02311 

.02642 

.02714 

.03055 

.031.52 

.0.3498 

.03625 

12 

13 

.02205 

.02317 

.026-49 

.02721 

.03063 

.03159  ! 

.03506 

03633 

13 

14 

.02271 

.02323 

.02655 

.02728 

.0.3070 

.03167 

.03514 

.0,3642 

14 

15 

.02277 

.02330 

.02662 

.02735 

.03077 

.03175 

.03521 

.03650 

15 

16 

.02283 

.02336 

.02669 

.02742 

.03084 

.03182 

.03529 

.0.36.58 

16 

17 

.02289 

.02343 

.02075 

.02749 

.03091 

.03190 

.035,37 

.03666 

17 

18 

.02295 

.02349 

.02682 

.02756 

.03098 

.03198 

.03544 

.0,3674 

18 

19 

.02302 

.02356 

.026S9 

.02763 

.03106 

.03205 

.03552 

.03683 

19 

20 

.02308 

.02362 

,02096 

.02770 

.03113 

.03213 

.03560 

.03691 

20 

21 

.02314 

.02369 

.02702 

.02777 

.03120 

.03221 

.03567 

.03699 

21 

22 

.02320 

.02375 

.02709 

.02784 

.03127 

.03228 

.03575 

.03708 

22 

23 

.02327 

.02382 

.02716 

.02791 

.03134 

.032.36 

.03583 

.03716 

23 

24 

.02333 

.02388 

.02722 

.02799 

.03142 

.03244 

.03590 

.03724 

24 

25 

.02339 

.02395  1 

.02729 

.02806 

.03149 

.03251 

.03598 

.03732 

25 

26 

.02345 

.02402  • 

.02736 

.02813 

.03156 

.032.59 

.0,3606 

.03741 

26 

27 

.023.52 

.02408 

.02743 

.02820 

.03163 

.03267 

.0.3614 

.03749 

27 

28 

.02358 

.02415 

.02749 

.02827 

.03171 

.03275 

.03621 

.037.58 

28 

20 

.02364 

.02421  1 

.027.56 

.02834 

.03178 

.03282 

.0.3629 

.03766 

29 

30 

.02370 

.02428  ! 

.02763 

.02842 

.03185 

.03290 

.03637 

.03774 

30 

31 

.02377 

.02435 

.02770 

.02849 

.03193 

.03298 

.0,3645 

.0,3783 

31 

32 

.02383 

.02441 

.02777 

.02856 

.03200 

.03306 

.03653 

.03791 

32 

33 

.02389 

.02448 

.02783 

.02863 

.0.3207 

.03313 

.03660 

.03799 

33 

34 

.02396 

.02454 

.02790 

.02870 

.03214 

.0.3321 

.03668 

.0.3808 

34 

35 

.02402 

.02461 

.02797 

.02878 

.03222 

.0.3329 

.03676 

.03816 

35 

36 

.02408 

.02468 

.02804 

.02885 

.03229 

.03337 

.03684 

.03825 

36 

37 

.02415 

.02474 

.02811 

.02892 

.03236 

.03345 

.0.3692 

.0,38.33 

37 

38 

.02421 

.02481 

.02818 

.02899 

.03244 

.0.3353 

.0,3699 

.03842 

38 

39 

.02427 

.024SS 

.02824 

.02907 

.0.3251 

.03360 

.03707 

.038,50 

39 

40 

.02434 

.02494 

.02831 

.02914 

.03258 

.03368 

.03715 

.03858 

40 

41 

.02440 

.02.501 

.02838 

.02921 

.03266 

.03376 

.03723 

.03867 

41 

42 

.02447 

.02508 

.02845 

.02928 

.03273 

.03384 

.0,3731 

.0,3875 

42 

43 

.024.53 

.02515 

.02852 

.02936 

.03281 

.0,3392 

.03739 

.03884 

43 

44 

.02459 

.02521 

.028.59 

.02943 

.03288 

.03400 

.03747 

.03892 

44 

45 

.02466 

.02528  1 

.02866 

.029.50 

.03295 

.03408 

.0,3754 

.0,3901 

45 

46 

.02472 

.02.535  1 

.0287'3 

.02958 

.03303 

.03416 

.0,3762 

.0,3909 

46 

47 

.02479 

.02542  1 

.02880 

.02965 

.0.3310 

.03424 

.0.3770 

.0,3918 

47 

48 

.0:M85 

.02548  1 

.02887 

.02972 

.03318 

.03432 

.03778 

.03927 

48 

49 

.02492 

.02555 

.02894 

.02980 

.03325 

.03439 

.03786 

.0,3935 

49 

50 

.02498 

.02.562 

.02900 

.02987 

.03333 

.03447 

.03794 

.03944 

50 

51 

.02504 

.02509 

.02907 

.02994 

.0.3.340 

.03455 

.0.3802 

.0,39.52 

51 

52 

.02511 

.02576 

.02914 

.03002 

.03347 

.0.3463 

.03810 

.03961 

52 

53 

.02517 

.02582 

.02921 

.03009 

.03355 

.0.3471 

.0.3818 

.0.3969 

53 

51 

.02524 

.02589 

.02928 

.03017 

.03362 

.0.3479 

.03826 

.0.3978 

.54 

55 

.02.530 

.02.596 

.02935 

.0.3024 

.03370 

.03487 

.0.3834 

.03987 

55 

56 

.02537 

.02603 

.02942 

.03032 

.03377 

.03495  j 

.03842 

.03995 

56 

57 

.02.543 

.02610 

.02949 

.03039 

.03385 

03503 

.0,38.50 

.04004 

57 

58 

.025.50 

.02617 

.029.56 

.03046 

.03392 

.03512 

.0,38.58 

.04013 

58 

50 

.02556 

.02624 

.02963 

.030.54 

.03400 

.03520 

.03866 

.(M021 

59 

60 

.02563 

.02630 

.02970 

.03061 

1  .03407 

.03528 

.03874 

.04030 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


r 

0 

16° 

17° 

18 

;° 

19° 

/ 

Vers. 

Exsec. 

Vers. 

1 
Exsec.  j 

Vers. 

Exsec. 

Vers. 

Exsec. 

.(m874 

.04080 

.04370 

.04569 

.04894 

.05146 

.05448 

.05762 

0 

1 

.(^82 

.04039 

.04378 

.04578 

.04903 

.05156 

.05458 

.05773 

1 

o 

.03S90 

.04047  1 

.04387 

.04588 

.04912 

.05166 

.05467 

.05783 

2 

3 

.03898 

.04056  j 

.04395 

.04597  ! 

.04921 

.05176 

.05477 

.05794 

3 

4 

.03906 

.04065 

.04404 

.04606 

.04930 

.05186 

.05486 

.05805 

4 

5 

.03914 

.04073 

.04412 

.046^6 

.04939 

.05196 

!  .05496 

.05815 

5 

6 

.03922 

.04082 

.04421 

.04625  ! 

.04948 

.05206 

i  .0.5505 

.05826 

6 

7 

.03930 

.04091 

.04429 

.04635  i 

j  .04957 

.05216 

1  .05545 

.05836 

7 

8 

.03938 

.04100 

.04438 

.04644 

.04907 

.05226 

1  .05524 

.05847 

8 

9 

.03946 

.04108 

.04446 

.04653 

.04976 

.05236 

.05534 

.05858 

9 

10 

.03954 

.04117 

.04455 

.04663 

.04985 

.05246 

.05543 

.05869 

10 

11 

.03903 

.04126 

.04464 

.04672 

.04994 

.05256  ' 

.05553 

.05879 

11 

V2 

.03971 

.04135 

.04472 

.04683 

.05003 

.05266 

.05562 

.05890 

12 

13 

.03979 

.04144 

.04481 

.04691 

.05012 

.05276 

.05572 

.05901 

13 

14 

.03987 

.04152 

.04489 

.04700 

.05021 

.05286 

.05582 

.05911 

14 

15 

.03995 

.04161 

.04498 

.04710 

.05030 

.05297 

.05591 

.05922 

15 

16 

.01003 

.04170 

.04507 

.04719 

.05039 

.05307 

.05601 

.05933 

16 

17 

.04011 

.04179  i 

.04515 

.04729  1 

.05048 

.05317 

.05610 

.05944 

17 

18 

.04019 

.04188  ' 

.04524 

.04738 

.05057 

.05327 

.05620 

.059,55 

18 

19 

.04028 

.04197 

.04533 

.04748  1 

.050G7 

.05337 

.05630 

.05965 

19 

20 

.04036 

.04206 

.04541 

.04757 

.05076 

.05347 

.05639 

.05976 

20 

21 

.04044 

.04214 

.04550 

.04767  ' 

.05085 

.05357 

.05649 

.05987 

21 

22 

.04052 

.04223 

.04559 

.04776  , 

.05094 

.05367 

.05658 

.0,5998 

22 

23 

.04060 

.04232 

.04567 

.04786 

.05103 

.05378 

.05668 

.06009 

23 

24 

.04069 

.04241 

.04576 

.04795 

.05112 

.05388 

.05678 

.06020 

24 

25 

.04077 

.04250  : 

.04585 

.04805  i 

.05122 

.05398 

.05687 

.06030 

25 

26 

.04085 

.04259 

.04593 

.04815  i 

.05131 

.05408 

.05697 

.06041 

26 

27 

.04093 

.04268 

.04602 

.04824  . 

.05140 

.05418 

.05707 

.060,52 

27 

28 

.04102 

.04^7 

.04611 

.04834 

.05149 

.05429 

.05716 

.06063 

28 

29 

.04110 

.04286 

.04620 

.04843  : 

.05158 

.05439 

.05726 

.06074 

29 

30 

.04118 

.04295 

.04628 

.04853 

.05168 

.05449 

.05736 

.06085 

30 

31 

.04126 

.04304 

.04637 

.04863 

.05177 

.05460  ' 

.05746 

.06096 

31 

32 

.04135 

.04313 

.04046 

.04872 

.05186 

.05470  ! 

.05755 

.06107 

32 

33 

.04143 

.04322 

.04055 

.04882 

.05195 

.05480  ' 

.05765 

.06118 

33 

34 

.04151 

.04331 

.04663 

.04891 

.05205 

.0.5490 

.05775 

.06129 

34 

35 

.04159 

.04:340 

.04673 

.04901  ' 

.05214 

.05501  ' 

.05785 

.06140 

35 

36 

.04168 

.04349 

.04681 

.04911 

.05223 

.05511  ' 

.05794 

.06151 

36 

37 

.04176 

.04358 

.04690 

.04920 

.05232 

.05521 

.05804 

.06162 

37 

38 

.04184 

.04367 

.04699 

.04930 

.05242 

.05532 

.05814 

.06173 

38 

39 

.04193 

.04376 

.04707 

.04940 

.05251 

.05542 

.05824 

.06184 

39 

40 

.04201 

.04385  j 

.04716 

.04950 

.05260 

.05552 

.05833 

.06195 

40 

41 

.04209 

.04394 

.04725 

.04959 

.05270 

.05563 

.05843 

.06206 

41 

42 

.04218 

.04403 

.04734 

.04969 

.05279 

.05573 

.05853 

.06217 

42 

43 

.04226 

.04413 

.04743 

.04979 

.05288 

.05584 

.0.5863 

.06228 

43 

41 

.04234 

.04422 

.04752 

.04989 

.05298 

.05594 

.05873 

.06239 

44 

45 

.04243 

.04431 

.04760 

.04998 

.05307 

.05604 

.05882 

.062.50 

45 

46 

04251 

.04440 

.04769 

.05008 

.05316 

.05015 

.05892 

.06261 

46 

47 

.04260 

.04449 

.04778 

.05018 

.05.326 

.05625 

.05902 

.06272 

47 

48 

.04268 

.04458 

.04787 

.05028 

.05335 

.0.5636 

.05912 

.06283 

48 

49 

.04276 

.04468 

.04796 

.05038 

.05344 

.05646 

.05922 

.06295 

49 

50 

.04285 

.04477 

.04805 

.05047 

.05354 

.05657 

.05932 

.06306 

50 

51 

.04293 

.04486  1 

.04814 

.05057 

.0.5363 

.05667 

.05942 

.06317 

51 

52 

.04302 

.04495  ' 

.04823 

.0.5067 

.05373 

.05078 

.05951 

.06328 

52 

53 

.04310 

.04504 

.04832 

.05077 

;  .05382 

.05688 

.0.5961 

.06339 

53 

54 

.04319 

.04514 

.04841 

.05087 

.05391 

.0,5699 

.05971 

.06,350 

54 

55 

.04327 

.04523 

.04850 

.05097 

.05401 

.0.5709 

.0,5981 

.06362 

55 

56 

.04336 

.04532 

.04858 

.05107 

.05410 

.05720 

.0.5991 

.06373 

56 

57 

.04344 

.04541  i 

.04867 

.05116 

.05420 

.05730 

.06001 

.06384 

57 

58 

.04353 

.04551 

.04876 

.05126 

.05429 

.0.5741 

.06011 

.06,^95 

58 

59 

.04361 

.04.560  I 

.04885 

.0.5136 

.0.5439 

.05751 

.06021 

.06407 

59 

60 

.04370 

.04569  ! 

.04894 

.05146  ; 

.05448 

.05762 

.06031 

.06418 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


337 


/ 

20" 

21° 

22° 

23° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.06031 

.06418 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

0 

1 

.06041 

.06429 

.01)652 

.07126 

.07293 

.07866 

.07961 

.08649 

1 

2 

.06051 

.06440 

.06663 

.07i:i8 

.07303 

.07879 

.07972 

.08663 

2 

3 

.06061 

.06452 

.06673 

.07150 

.07314 

.07892 

.07984 

.08676 

3 

4 

.06071 

.00463 

.06684 

.07162 

.07325 

.07904 

.07995 

.08690 

4 

5 

.06081 

.06474 

.06691 

.07174 

.07336 

.07917 

.08006 

.08703 

5 

6 

.06091 

.06486 

.06705 

.07186 

.07347 

.07930 

.08018 

.08717 

6 

7 

.06101 

.06497 

.06715 

.07199 

.07358 

07043 

.08029 

.08730 

7 

8 

.06111 

.06508 

.06726 

.07211 

.07369 

.07955 

.08041 

.08714 

8 

9 

.06121 

.06520 

.00736 

.07223 

.07380 

.07968 

.08052 

.08757 

9 

10 

.06131 

.06531 

.06747 

.07235 

.07391 

.07981 

.08064 

.08771 

10 

11 

.06141 

.06542 

.06757 

.07247 

.07402 

.07994 

.08075 

.08784 

11 

l:i 

.06151 

.00554 

.06768 

.07259 

.07413 

.08006 

.1)8086 

.08798 

12 

13 

.06161 

.06565 

.06778 

.07271 

.07424 

.08019 

.08098 

.08811 

13 

14 

.06171 

.06577 

.06789 

.07283 

.07435 

.08032 

.08109 

.08825 

14 

15 

.06181 

.06588 

.06799 

.07295 

.07446 

.08045 

.08121 

.08839 

15 

16 

.06191 

.06000 

.06810 

.07307 

.07457 

.08058 

.08132 

.08852 

16 

17 

.06201 

.06611 

.06820 

.07320  ! 

.07468 

.08071 

.08144 

.08866 

17 

18 

.06211 

.06622 

.06831 

.07332 

.07479 

.08084 

.08155 

.08880 

18 

19 

.06221 

.06634 

.06841 

.07344 

.07490 

.08097 

.08167 

.08893 

19 

20 

.06231 

.06645 

.06852 

.07356 

.07501 

.08109 

.08178 

.08907 

20 

21 

.06241 

.06657 

.06863 

.07368 

.07512 

.08122 

.08190 

.08921 

21 

22 

.06252 

.06668 

.06873 

.07380 

.07523 

.08135 

.08201 

.08934 

22 

23 

.06263 

.06680 

.06884 

.07393  i 

.07534 

.08148 

.08213 

.08948 

23 

24 

.06272 

.06691 

.06894 

.07405  1 

.07545 

.08161 

.08225 

.08962 

24 

25 

.06282 

.06703' 

.06905 

.07417  1 

.07556 

.08174 

.08236 

.08975 

25 

26 

.06292 

.06715 

.06916 

.07429  ; 

.07568 

.08187 

.08248 

.08989 

26 

27 

.06302 

.06726 

.06926 

.07142  i 

.07579 

.08200 

.08259 

.09003 

27 

28 

.06312 

.06738 

.06937 

.07454 

.07590 

.08213 

.08271 

.09017 

28 

29 

.06323 

.06749 

.06948 

.07466 

.07601 

.08226 

.08282 

.09030 

29 

30 

.06333 

.06761 

.06958 

.07479 

.07612 

.08239 

.08294 

.09044 

30 

31 

.06343 

.06773 

.06969 

.07491 

.07623 

.08252 

.08306 

.09058 

31 

32 

.06353 

.06784 

.06980 

.07503 

.07634 

.08265 

.08317 

.09072 

32 

33 

.06363 

.06796 

.06990 

.07516 

.07645 

.08278 

.08329 

.09086 

33 

34 

.06374 

.06807 

,07001 

.07528 

.07657 

.08291 

.08340 

.09099 

34 

35 

.06384 

.06819 

.07012 

.07540 

.07668 

.08305 

.08352 

.09113 

35 

36 

.06394 

.06831 

.07022 

.07553 

.07679 

.08318 

.08364 

.09127 

36 

37 

.06404 

.06843 

.07033 

.07565 

.07690 

.08331 

.08375 

.09141 

37 

38 

.06415 

.06854  i 

.07044 

.07578  ' 

.07701 

.08344 

.08387 

.09155 

38 

39 

.06425 

.06866  1 

.070.55 

.07590  , 

.07713 

.08357 

.08399 

.09169 

39 

40 

.06435 

.06878  j 

.07065 

.07602 

.07724 

.08370 

.08410 

.09183 

40 

41 

.06445 

.06889 

.07076 

.07615 

.07735 

..08383 

.08422 

.09197 

41 

42 

.06456 

.06901  1 

.07087 

.07627  ; 

.07746 

.08397 

.08434 

.09211 

42 

43 

.06466 

.06913  ! 

.07098 

.07640 

.07757 

.08410 

.08445 

.09224 

43 

44 

.06476 

.06925 

.07108 

.07652  i 

.07769 

.08423 

.08457 

.09238 

44 

45 

.06486 

.06936 

.07119 

.07665  ! 

.07780 

.08436 

.08469 

.09252 

45 

46 

.06497 

.06948 

.07130 

.07677  1 

.07791 

.08449  I 

.08481 

.09266 

46 

47 

.06507 

.06960 

.07141 

.07690 

.07802 

.08463 

.08492 

.09280 

47 

48 

.06517 

.06972 

.07151 

.07702 

.07814 

.08476 

.08504 

.09294 

48 

40 

.06528 

.06984 

.07162 

.07715 

.07825 

.08489 

.08516 

.09308 

49 

50 

.06538 

.06995  j 

.07173 

.07727 

.07836 

.08503 

.08528 

.09323 

50 

51 

.06548 

.07007 

.07184 

.07740 

.07848 

.08516 

.08539 

.09337 

51 

52 

.065.59 

,07019 

.07195 

.07752 

.07859 

.08529 

.08551 

.09351 

52 

53 

.06569 

.07031 

.07206 

.07765 

,  .07870 

.08.542 

.08563 

.09365 

53 

54 

.06580 

.07043 

.07216 

.07778 

.07881 

.08556 

.08575 

.09379 

54 

^  55 

.06590 

.07055 

.07227 

.07790 

.07893 

.08569 

.08.586 

.09393 

55 

56 

.06600 

.070()7 

.07238 

.07803 

.07904 

.08582 

.08598 

.09407 

56 

57 

.06611 

.07079 

.07^9 

.07816 

!  .07915 

.08596 

.08610 

.09421 

57 

58 

.06621 

.07091  ' 

.07260 

.07828 

1  .07927 

.081)09 

.08622 

.09435 

58 

59 

.06632 

.07103  1 

.07271 

.07841 

.07938 

.08623  i 

.086:i4 

.09449 

59 

60 

.06642 

.07115  ! 

.07282 

.07853 

.079.50 

.08636  1 

.08(i45 

.094W 

60 

338 


TABLE   Xm.— VERSINES  AND   EXSECANTS. 


§ 

24- 

25° 

26° 

27° 

0 

w 

Vers. 

1 

Exsec. 

.09404 

Vers. 
.09300 

Exsec. 

Vers. 

Exsec.  j 

Vers. 

Exsec, 

0  1 

.08645 

.10:538 

.10121 

.11260 

.10899 

.122.33 

1  1 

.08657 

.09478 

.09382 

.10:353 

.10133 

.11276  ; 

.10913 

.12249 

1 

.08669 

.09492 

.09394 

.10368 

.10146 

.11292  ! 

.10926 

.12266 

2 

3 

.a8681 

.09506 

.09406 

.10383  ; 

.10159 

.11:308 

.10939 

.12283 

3 

4 

.08693  i 

.09520 

.09418 

.10:398 

.10172 

.11:323 

.10952 

.12299 

4 

5 

.08705 

.09535 

.09431 

.10413 

.10184 

.113:39 

.10965 

.12:316 

5 

6 

.08717 

.09549 

.09443 

.10428 

.10197 

.11:355  ! 

.10979 

.12:3a3 

6 

7  ! 

.08728 

.09563 

.09455 

.10443  , 

.10210 

.11:371 

.10992 

.12349 

7 

8  1 

.08740 

.09577 

.09468 

.10458  1 

.10223 

.11387  ; 

.11005 

.12366 

8 

9 

.08752 

.09592 

.09480 

.10473  ; 

.102:36 

.11403 

.11019 

.12383 

9 

10 

.08764 

.09606 

.09493 

.10488  ' 

.10248 

.11419 

.11032 

.12400 

10 

11 

.08776 

.09020 

.09505 

.10503 

.10261 

.11435 

.11045 

.12416 

11 

12 

.08788 

.09635 

.09517 

.10518 

.10274 

.11451 

.11058 

.1243:3 

12 

13 

.08800 

.09649 

.09530 

.  105:33 

.10287 

.11467 

.11072 

.12450 

13 

14 

.08812 

.09663 

.09542 

.10549  1 

.10300 

.11483 

.11085 

.12467 

14 

15 

.08824 

.09678  \ 

.09554 

.10564 

.10:313 

.11499 

.11098 

.12484 

15  ' 

16 

.08836 

.09692 

.09567 

.10579 

.10326 

.11515 

1  .11112 

.12501 

16 

17 

.08848 

.09707 

.09579 

.10594 

.10338 

.11531 

'  .11125 

.12518 

17 

18 

.08860 

.09721 

.09592 

.10609 

.10351 

.11547 

.11138 

.12534 

18 

19 

.08872 

.097.35 

.09604 

.10625 

.10364 

.11.563  i 

.11152 

.12.5.51 

19 

20 

.08884 

.09750 

.09617 

.10640  1 

. 10377 

.11579 

.11165 

.12568 

20 

21 

.08896 

.09764 

.09629 

.10655 

.10.390 

.11595 

.11178 

.12585 

21 

22 

.08908 

.09779 

.09042 

.10070 

.10403 

.11611 

.11192 

.12602 

22 

23 

.08920 

.09793 

.09654 

.10686 

.10416 

.11627 

.11205 

.12619 

23 

24 

.08932 

.09808 

.09666 

.10701  ! 

.10429 

.11643 

.11218 

.126.36 

24 

25 

.08944 

.09822 

.09679 

.10716  ' 

.10442 

.11659 

.112:32 

.12653 

25 

26 

.08956 

.09837 

.09691 

.10731 

.10455 

.11675 

.11245 

.12670 

26 

27 

.08968 

.09851 

.09704 

.10747 

.10468 

.11691 

.11259 

.12687 

27 

28 

.03980 

.09866 

.09716 

.10762 

.10481 

.11708 

.11272 

.12704 

28 

29 

.08992 

.09880 

.09729 

.10777  i 

.10494 

.11724 

.11285 

.12721 

29 

30 

.09004 

.09895  ; 

.09741 

.10793 

.10507 

.11740 

.11299 

.12738 

30 

31 

.09016 

.09909 

.09754 

.10808 

.10520 

.11756 

.11312 

.12755 

31 

32 

.09028 

.09924 

.09767 

.10824 

.10533 

.11772 

.11326 

.12772 

32 

33 

.09040 

.09939 

.09779 

. 10839 

.10546 

.11789 

.11339 

.12789 

33 

34 

.09052 

.099.53 

.09792 

.10854 

.10559 

.11805 

.11353 

.12807 

34 

35 

.09064 

.09908 

.09804 

.10870 

.10572 

.11821 

.11:366 

.12824 

35 

36 

.09076 

.09982 

.09817 

.10885 

.10585 

.11838 

.11380 

.12841 

36 

37 

.09089 

.09997 

.09829 

.10901 

.10598 

.11854 

.11393 

.12858 

37 

38 

.09101 

.10012 

.09842 

.10916 

.10611 

.11870 

.11407 

.12875 

38 

39 

.09113 

.10026 

.09854 

.109:32 

.10624 

.11880 

.11420 

.12892 

39 

40 

.09125 

.10041 

.09867 

.10947 

.10637 

.11903 

.11434 

.12910 

10 

41 

.09137 

.10055 

.09880 

.10963 

.10650 

.11919 

.11447 

.12927 

41 

42 

.09149 

.10071 

1  .09892 

.10978 

.10663 

.119:36 

.11461 

.12944 

42 

43 

.09101 

.10085 

1  .09905 

.10994  ' 

.10676 

.11952 

.11474 

.12961 

43 

44 

.09174 

.10100 

'  .09918 

.11009 

.10689 

.11968 

.11488 

.12979 

44 

45 

.09186 

.10115 

.09930 

.11025 

.10702 

.11985 

.11501 

.12996 

4,5 

46 

.09198 

.10130 

.09943 

.11041 

.10715 

.12(X)1 

.11515 

.1:3013 

46 

47 

.09210 

.10144 

.09955 

.11056 

.10728 

.12018 

.11528 

.13031 

47 

48 

.09222 

.10159 

.09968 

.11072 

.10741 

.12034 

.11542 

.13048 

48 

49 

.09234 

.10174 

.09981 

.11087 

.10755 

.12051 

.11555 

.13065 

49 

50 

.09247 

.10189 

.09993 

.11103 

.10768 

.12067 

.11569 

.13083 

50 

51 

.09259 

.10204 

.10000 

.11119 

.10781 

.12084 

.11583 

.13100 

51 

52 

.09271 

.10218 

.10019 

.111:34 

.10794 

.12100 

!  .11596 

.13117 

52 

53 

.09283 

.10233 

.10032 

.11150 

;  .10807 

.12117 

.11610 

.13135 

53 

54 

.09296 

.10248 

.10044 

.11166 

.10820 

.  121:33 

.11623 

.13152 

.54 

55 

.09308 

.10263 

.10057 

.11181 

.10833 

.12150 

.11037 

.13170 

55 

56 

.09320 

.1027'8 

.10070 

.11197 

.10847 

.12166 

.11651 

.13187 

56 

57 

.093.32 

.10293 

. 10082 

.11213 

.10860 

.12183 

.11664 

.1:3205 

57 

58 

.09345 

.10308 

.10095 

.11229 

.10*^73 

.12199 

.11678 

.13222 

58 

59 

.003.i7 

.10323 

.10108 

.11244 

.10886 

.12216 

.11(592 

.13240 

59 

60 

.09369 

1  .10338 

.10121 

.11260 

.10899 

.12233 

.11705 

.13257 

60 

TABLE   XIII.-VERSINES  AND  EXSECANTS. 


:J39 


/ 

28° 

29° 

30° 

31» 

/ 
0 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.11705 

.13257 

.12538 

.14335 

.13397 

.15470 

.14283 

.10003 

1 

.11719 

.13275 

.12552 

.14354 

.13412 

.15489 

.14298 

.10084 

1 

2 

.11733 

.13292 

.12506 

.14372 

.r5427 

.15509 

.14313 

.107'04 

2 

3 

.11746 

.13310 

.12580 

.14391 

.J.3441 

,15528 

.14328 

.10725 

3 

4 

.11700 

.13327 

.12595 

.14409 

.13456 

. 15548 

.14343 

.16745 

4 

5 

.11774 

.13345 

.12609 

.14428 

.1^70 

. 15567 

.14358 

.16766 

5 

6 

.11787 

.13362 

.12023 

.14446 

.13485 

.15587 

.14373 

.16786 

6 

7 

.11801 

.13380 

.12037 

.14405 

.13499 

.15606 

.14388 

.10800 

7 

8 

.11815 

.13398 

.12651 

.14483 

.13514 

.15026 

.14403 

: 10827 

8 

9 

.11828 

.13415 

.12065 

.14502 

.13529 

.15045 

.14418 

.10848 

9 

10 

.11342 

.13133 

.12679 

.14521 

.13543 

.15065 

.14433 

.10808 

10 

11 

.11856 

.13451 

.12694 

.14539 

.13.558 

.15684 

.14449 

.10889 

11 

12 

.11870 

.i;«68 

.12708 

.14558 

.13573 

.15704 

.14404 

.10909 

12 

13 

.11883 

.13486 

.12722 

.14576 

.13.587 

.15724 

.14479 

.16930 

13 

14 

.11897 

.13504 

.12736 

.14595 

.13602 

.15743 

.14494 

.16950 

14 

15 

.11911 

.13521 

.12750 

.14014 

.13010 

.15763 

.14509 

.16971 

15 

16 

.11925 

.13539 

.12705 

.14632 

.13031 

.15782 

.14524 

.10992 

16 

17 

.11938 

.13557 

.12779 

.14651 

.13646 

.15802 

.14539 

.17012 

17 

18 

.11952 

.13.575 

.12793 

.14070 

.13600 

.15822 

.14554 

.17033 

18 

19 

.]1%6 

.13593 

.12807 

.14689 

.13075 

.15841 

.14569 

.17054 

19 

20 

.11980 

.13610 

.12822 

.14707 

.13090 

.15801 

.14584 

.17075 

20 

21 

.11994 

.13628 

.12836 

.14726 

.13705 

.15881 

.14599 

.17095 

21 

23 

.12007 

.13646 

.12850 

.14745 

.13719 

.15901 

.14615 

.17116 

22 

23 

.12021 

.13664 

.12804 

.14704 

.13734 

.15920 

.14630 

.17137 

23 

24 

.12035 

.13682 

.1287'9 

.14782 

.13749 

.15940 

.14645 

.17158 

24 

25 

.12049 

.13700 

.12893 

.14801 

.13703 

.15960 

.14660 

.17178 

25 

20 

.12063 

.13718 

.12907 

.14820 

.13778 

.15980 

.14675 

.17199 

26 

27 

.12077 

.13735 

.12921 

.14839 

.13793 

.16000 

.14690 

.17220 

27 

28 

.12091 

.13753 

.12936 

.14858 

.13808 

.16019 

.14706 

.17241 

28 

29 

.12104 

.13771 

.12950 

.14877 

.13822 

.10039 

.14721 

.17262 

29 

30 

.12118 

.13789 

.12904 

.14896 

.13837 

.16059 

.14736 

.17283 

30 

31 

.12132 

.13807 

.12079 

.14914 

.13852 

.16079 

.14751 

.17304 

31 

32 

.12146 

.13825 

.12993 

.14933 

.13807 

.16099 

.14766 

.17325 

32 

33 

.12160 

.13843 

.13007 

.1495'> 

.13881 

.16119 

.14782 

.17346 

33 

34 

.12174 

.13861 

.13022 

.14971 

.13896 

.16139 

.14797 

.1.7307 

34 

35 

.12188 

.13879 

.13036 

.14990 

.13911 

.16159 

.14812 

.17388 

35 

36 

.12202 

.13897 

.13051 

.15009 

.13926 

.16179 

.14827 

.17409 

36 

37 

.12216 

.13916 

.13005 

.15028 

.13941 

.16199 

.14843 

.17430 

37 

38 

.12230 

.139:34 

.13079 

.15047 

.13955 

.16219 

.14858 

.17451 

38 

39 

.12244 

.13952 

.13094 

.15006 

.13970 

.10239 

.14873 

.17472 

39 

40 

.12257 

.13970 

.13108 

.15085 

.13985 

.16259 

.14888 

.17493 

40 

41 

.12271 

.13988 

.13122 

.15105 

.14000 

.16279 

.14904 

.17514 

41 

42 

.12285 

.14006 

.13137 

.15124 

.14015 

.16299 

.14919 

.17535 

42 

43 

.12299 

.14024 

.131.51 

.15143 

.14030 

.16319 

.14934 

.17556 

43 

44 

.12313 

.14042 

.13106 

.15102 

.14044 

.16339 

.14949 

.17577 

44 

45 

.12327 

.14061 

.13180 

.15181 

.14059 

.16359 

.14965 

.17598 

45 

46 

.12:^41 

.14079 

.13195 

.15200 

.14074 

.16380 

.14980 

.17020 

46 

47 

.12355 

.14097 

.13209 

.15219 

.14089 

.16400 

.14995 

.17641 

47 

48 

.12309 

.14115 

.13223 

.15239 

.14104 

.16420 

.15011 

. 17062 

48 

49 

.12383 

.14134 

.13238 

.15258 

.14119 

.16440 

.15026 

.17083 

49 

50 

.12397 

.14152 

.132.52 

.15277 

.14134 

.16460 

.15041 

.17704 

50 

51 

.12411 

.14170 

.13207 

.15296 

.14149 

.16481 

.15057 

.17726 

51 

52 

.12425 

.14188 

.13281 

.15315 

.14104 

.16501 

.15072 

.17747 

52 

53 

.12439 

.14207 

.13296 

.15335 

.14179 

.16521 

.15087 

.17768 

53 

54 

.12454 

.14225 

.13310 

.15354 

.14194 

.16.541 

.15103 

.17790 

54 

55 

.12408 

.14243 

.13325 

.15373 

.14208 

.16562 

.15118 

.17811 

55 

56 

.12482 

.14262 

.13339 

.15393 

.14223 

.16582  1 

.15134 

.17832 

56 

57 

.12490 

.14280 

.13354 

.15412 

.14238 

.16602  1 

.15149 

.17854 

57 

58 

.12.510 

.14299 

.13368 

.15431 

.14253 

.16623  1 

.15164 

.17875 

58 

59 

12524 

.14317 

.13383 

.154.51 

.14268 

.16043 

.15180 

.17890 

59 

60 

.12538 

.14335 

A^m 

.15470 

.14283 

.16663 

.15195 

.17918 

60 

uo 


TABLE  XIII.-VERSINES  AND   EXSECANTS. 


0 

32" 

33° 

84° 

35° 

J 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

/ 

.15195 

.17918 

.16133 

.19236 

.17096 

.20622 

.18085 

.22077 

0 

1 

.15211 

.17939 

.16149 

.1.9259 

.17113 

.20645 

.18101 

.22102 

1 

2 

.15226 

.17961 

.16165 

.19281 

.17129 

.20669 

.18118 

.22127 

2 

3 

.15241 

.17982 

.16181 

.19304 

.17145 

.20693 

.18135 

.22152 

3 

4 

.15257 

.18004 

.16196 

.19327 

.17161 

.20717 

.18152 

.22177 

4 

5 

.15272 

.18025 

.16212 

.19349 

.17178 

.20740 

.18168 

.22202 

5 

6 

.15288 

.18047 

.16228 

.19372 

.17194 

.20764 

.18185 

.22227 

6 

7 

.15303. 

.18068 

.16244 

.19394 

.17210 

.20788 

.18202 

.22252 

7 

8 

.15319 

.18090 

.16260 

.19417 

.17227 

.20812 

.18218 

.22277 

8 

9 

.15334 

.18111 

.16276 

.19440 

.17243 

.20836 

.18235 

.22302 

9 

10 

.15350 

.18133 

.16292 

.19463 

.17259 

.20859 

.18253 

.22327 

10 

11 

.15365 

.18155 

.16308 

.19485 

.17276 

.20883 

.18269 

.22352 

11 

12 

.15381 

.18176 

.16324 

.19508 

.17292 

.20907 

.18286 

.22377 

12 

13 

.15396 

.18198 

.16340 

.19531 

.17308 

.20931 

.18302 

.22402 

13 

14 

.15412 

.18220 

.16355 

.19554 

.17325 

.20955 

.183^9 

.22428 

14 

15 

.15427 

.18241 

.16371 

.19576 

1  .17341 

.20979 

.18336 

.22453 

15 

16 

.15443 

.18263 

.16387 

.19599 

.17357 

.21003 

.18:353 

.22478 

16 

17 

.15458 

.18285 

.16403 

.19622 

.17374 

.21027 

.18369 

.22503 

17 

18 

.15474 

.18307 

.16419 

.19645 

.17390 

.21051 

.18386 

.22528 

18 

19 

.15489 

.18328 

.16435 

.19668 

.17407 

.21075 

.18403 

.22554 

19 

20 

.15505 

.18350 

.16451 

.19691 

.17423 

.21099 

.18420 

.22579 

20 

21 

.15520 

.18372 

.10467 

.19713 

.17439 

.21123 

.18437 

.22604 

21 

22 

.15536 

.18394 

.16483 

.19736 

.17456 

.21147 

1  .18454 

.22629 

22 

23 

.15552 

.18416 

.16499 

.19759 

.17472 

.21171 

[  .18470 

.22655 

23 

24 

.15567 

.18437 

.16515 

.i':'782  ; 

.17489 

.21195 

1  .18187 

.22680 

24 

25 

.1558;3 

.184.59 

.16531 

.19805  1 

1  .17505 

.21220 

.18504 

.22706 

25 

26 

.15598 

.18481 

.16547 

.19828 

.17522 

.21244 

.18521 

.22731 

26 

27 

.15614 

.18.503 

.16503 

.19851 

.17538 

.21208 

.185.38 

.22756 

27 

28 

.15630 

.18525 

.16579 

.19874 

.17554 

.21292 

.18555 

.22782 

28 

29 

.15645 

.18547 

.16595 

.19897 

.17571 

.21316 

.18572 

.22807 

29 

30 

.15661 

.18569  ; 

.16611 

.19920 

.17587 

.21341 

.18588 

.22833 

30 

31 

.15676 

.18591 

.16627 

.19944 

.17604 

.21365 

.18605 

.22858 

31 

32 

.15692 

.18613 

.16644 

.19967 

.17620 

.21389 

.18622 

.22884 

32 

33 

.15708 

.18635 

.16660 

.19990 

.17637 

.21414 

.18639 

.22909 

33 

34 

.15723  1 

.15739  ! 

.18657 

.16676 

.20013 

.17653 

.21438 

.18656 

.22935 

34 

35 

.18679 

.16692 

.20036 

.17670 

.21462 

.18073 

.22960 

35 

36 

.15755 

.18701 

.16708 

.20059 

.17686 

.21487 

.18690 

.22986 

36 

37 

.15  .TO 

.18723 

.16724 

.200&3  1 

.17703 

.21511 

.18707 

.23012 

37 

38 

.15780 

.18745 

.16740  : 

.20106 

.17719 

.21535 

.18724 

.23037 

38 

39 

.15802 

.18767 

.16756  i 

.20129 

.17736 

.21560  i 

.18741 

.23063 

39 

40 

.15818 

. 18790 

.16773  1 

.20153 

.17752 

.21584 

.18758 

.23089 

40 

41 

.15833 

.18812 

.16788  i 

.20176 

.17769 

.21609  : 

.18775 

.23114 

41 

42 

.15849 

.18834 

.16805 

.20199 

.17786 

.21633 

.18792 

.23140 

42 

43 

.1.5865 

.18856 

.16821 

.20222 

.17802 

.21658 

.18809 

.23166 

43 

44 

.15880 

.18878 

.16837  1 

.20246 

.17819 

.21682 

.18826 

.23192 

44 

45 

. 15896 

.18901 

.16853 

.20269 

.17835 

.21707  i 

.18843 

.23217 

45 

46 

.15912 

.18923 

.16869 

.20292 

.17852 

.21731  s 

.18860 

.23243 

46 

47 

.1592S 

.18945 

.16885 

.20316 

.17868 

.21756 

.18877 

.23269 

47 

48 

.15943 

.18967 

.16902  1 

.20339  ! 

.17885 

.21781 

.18894 

.23295 

48 

49 

.15959 

.18990 

.16918 

.20363 

.17902 

.21805 

.18911 

.23321 

49 

50 

.15975 

.19013 

.16934 

.20386 

.17918 

.21830 

.18928 

.23347 

50 

51 

.15991 

.19034 

.16950 

.20410 

.17935 

.218.55 

.18945 

.23373 

51 

52 

.1G006 

.19057 

.16906 

.20433 

.17952 

.21879 

.18962 

.23399 

52 

53 

.16022 

.19079 

.16983 

.20457 

.17968 

.21904 

.18979 

.23424 

53 

54 

.10038 

.19102 

.16999 

.20480 

.17985 

.21929 

.18996 

.23450 

54 

55 

.16054 

.19124  : 

.17015 

.20504 

.18001 

.21953 

.19013 

.23476 

55 

56 

.16070 

.19146 

.17031 

.20.527 

.18018 

.21978 

.19030 

.23502 

56 

57 

.16085 

.19109  j 

.17047 

.20551  i 

.18035 

.23003 

.19047 

.23529 

57 

58 

.16101 

.19191 

.17064 

.20575  ' 

.18051 

.22028 

.19064  ! 

.23555 

58 

59 

.16117 

.19214 

.17080 

.20598 

.18068 

.220.53 

.19081 

.23581 

59 

60 

.16133  i 

.19236 

.17096 

.20622 

.18085 

.22077 

.19098 

.23607 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


341 


36" 


Vers. 


Exsec. 


37° 


Vers. 


Exsec. 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
U 
15 
16 
17 
18 
19 
20 

21 

23 
24 
25 
20 
27 
28 
29 
30 

3t 

32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


52 
53 

55 
56 
57 
58 
59 
60 


.19098 
.19115 
.19133 
.19150 
.19167 
.1918! 
.19201 
.19218 
.19235 
.19252 
.19270 

.19287 
.19304 
.19321 
.19:338 
.19356 
.19373 
.19390 
.19407 
.19424 
.19442 

.19459 
.19476 
.19493 
.19511 
.19528 
.19545 
.19502 
.19580 
.19597 
.19614 

.19632 
.19649 
.19666 
.1'J684 
.19701 
.19718 
.19736 
.197.53 
.19770 
.19788 

.19805 
.19822 
.19840 
.198.57 
.19875 
.19892 
.19909 
.19927 
.19944 
.19962 

.19979 
.19997 
.20014 
.20032 
.20049 
.200(56 
.20084 
.20101 
.20119 
.20136 


.23007 
.2:3633 
.23659 
.23685 
.23711 
.2:37:38 
.23764 
.2:3790 
.2:3816 
.2:3843 
.23869 

.23895 
.23922 
.23948 
.23975 
.24001 
.24028 
.24054 
.24081 
.24107 
.24134 

.24160 
.24187 
.24213 
.^i^O 
.24267 
.24293 
.24320 
.24347 
.24:373 
.24400 

.24427 
.24454 
.24481 
.24508 
.:215;34 
.21561 
.24588 
.24615 
.24&42 
.24669 

.24696 
.24723 
.24750 
.24777 
.24804 
.248:32 
.24859 
.24880 
.24913 
.21940 

.24967 
.j^995 
.25022 
.25049 
.2.5077 
.25104 
.25131 
.25159 
.25186 
.25214 


.201. SO 
.20154 
.20171 
.20189 
.20207 
.20224 
.20242 
.20259 
.20277 
.20294 
.20312 

.20329 
.20347 
.20365 
.20382 
.20400 
.20417 
.20435 
.20453 
.20470 
.20488 

.20506 
.20523 
.20541 
.20559 
.20576 
.20594 
.20612 
.20629 
.20647 
.20665 

.20682 
.20700 
.20718 
.20736 
.20753 
.20771 
.20789 
.20807 
.20824 
.20842 

.20860 
.20878 
.20895 
.20913 
.20931 
.20949 
.20967 
.20985 
.21002 
.21020 

.21038 
.21056 
.21074 
.21092 
.21109 
.21127 
.21145 
.21163 
.21181 
.21199 


.2.-^214 

! 25269  I 
.25296  i 
.25324 
.25351  ; 
.25379 
.25406 
.25434 
.25462  ! 
.25489  I 

.25517 
.25545 
.25572  ; 
.25600 
.25628  ! 
.25656  i 
.25683 
.25711  j 
.25739 
.25767 

.25795 

.25823 

.25851 

.25879 

.25907 

.25935  I 

.25963 

.25991 

.26019 

.26047  I 

.26075 

.26104 

.26132 

.26160 

.26188 

.26216 

.26245 

.26273  r 

.26301 

.26330 

.26.3.58 

.263G7 

.26415  i 

.26443 

.26472 

.26500 

.26529 

.26557 

.26586 

.26615 

.26643  I 

.26672 

.26701 

.26729 

.26758 

.26787 

.26815 

.26844 

.26873  : 

.26902  I 


38» 

3 

d° 

Vers. 

Exsec. 

Vers. 

Exsec. 

:  .21199 

.26902 

.22285 

.2;-676 

.21217 

.26931 

.22:304 

.28706 

:  .212:35 

.26960 

.22:322 

.28737 

.212.53 

.26988 

.22340 

.28767 

.21271 

.27017 

.22359 

.28797 

.21289 

.27046 

.22377 

.28823 

.21307 

.27075 

.22395 

.28858 

.21.324 

.27104 

.22414 

.28889 

.21.342 

.271:33 

.22432 

.28919 

.21 300 

.27162 

.22450 

.289.50 

.21378 

.27191  t 

.22469 

.28980 

.21390 

.27221 

.22487 

.29011 

.21414 

.27250 

.22506 

.29042 

.21432 

.27279 

.22524 

.29072 

.21450 

.273aS 

.22542 

.29103 

.21468 

.27337 

.22501 

.29133 

.21486 

.27366 

.22579 

.29164 

.21504 

.27.396 

.22598 

.29195 

.21522 

.27425 

.22616 

.29226 

.21540 

.274.>1 

.22634 

.29256 

.21558 

.2748:3 

.22653 

.29287 

.21576 

.27513 

.22671 

.29318 

.21595 

.27^2  i 

.22690 

.29349 

.21613 

.27572 

.22708 

.29380 

.21631 

.27601 

.22727 

.29411 

.21649 

.27630 

.22745 

.29442 

.21667 

.27660  1 

.22764 

.29473 

.21685 

.27689 

.22782 

.29504 

.21703 

.27719 

.22801 

.29.535 

.21721 

.27748 

.22819 

.29506 

.21739 

.27778 

.22838 

.29597 

.21757 

.27807 

.22856 

.29628 

.21775 

.278:37 

.22875 

.29659 

.21794 

.27867 

.22893 

.29690 

.21812 

.27896 

.22912 

.29721 

.218.30 

.27926 

.22930 

.29752 

.21848 

.27956 

.22949 

.29784 

.21866 

.27985 

.22967 

.29815 

.21884 

.28015 

.22986 

.29^46 

.21902 

.28045 

.2.3004 

.29877 

.21921 

.28075  ' 

.23023 

.29909 

.21939 

.28105 

.23041 

.29940 

.21957 

.281:34 

.23060 

.29971 

.21975 

.28164 

.23079 

.30003 

.21993 

.28194 

.23097 

.300.34 

.22012 

.282^ 

.23116 

.30006 

.22030 

.28254 

.23134 

.3lX)97 

.22048 

.28284 

.23153 

.30129 

.22006 

.28314 

.23172 

.30160 

.22084 

.28344 

.2:3190 

.30192 

.22103 

.28374 

.23209 

.30223 

.22121 

.28404 

.23228 

..30255 

.22139 

.28434 

.23246 

.;30287 

.22157 

.28464 

.23205 

.30318 

.22176 

.28495 

.2:3283 

.30a50 

.22194 

.28525 

.2.3302 

.30382 

.22212 

.285.55 

.23321 

.30413 

.22231 

.28585 

.23.3.39 

.30445 

.22249 

.28615 

.2:3.3.58 

..30477 

.22267 

.28646 

.2:3.377 

.;30509 

.22285 

.28076 

.23396 

.30.541 

o  10 


Table  xiii.— versines  and  exsecakts. 


/ 

40» 

41» 

42° 

43'» 

f 

Vers. 

Kxsec. 

Vers. 

Exsec.  1 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.23:396 

.30541 

•  .24529 

.32501 

.25686 

.34.563 

.26865 

.367^3 

0 

1 

.2:3414 

.30573 

.24548 

.32535 

.25705 

.34599  ' 

.26884 

.36770 

1 

2 

.2:34:33 

.30605 

.24567 

..32568  1 

.25724 

.34634  ! 

.26904 

.36807 

2 

3 

.2^452 

.30636 

.24556 

.32C02 

.2.5744 

.34669 

.26924 

.36844 

3 

4 

.2^470 

.30668 

.24605 

.:32G:36 

.25763 

.34704 

.26944 

..36881 

4 

5 

.2^489 

.30700 

.24625 

.32669 

.2.5783 

.34740 

.26964 

.36919 

5 

6 

.23508 

.307:32 

.24644 

.32703 

.25802 

.34:75 

.26984 

.36956 

6 

7 

.2:3527 

.30764 

.24663 

.327:37 

.25822 

.34811 

.27004 

.36993 

7 

8 

.23545 

.30796 

.24682 

..32770  i 

.25841 

.34846 

.27024 

.37030 

8 

9 

.23564 

.308-29 

.24701 

.32804  i 

.25861 

.34882 

.27043 

.37068 

9 

10 

.23583 

.30861 

.24720 

.32838  ' 

.25880 

.34917 

.27063 

.37105 

10 

11 

.23603 

.30893 

.24739 

..32872 

.25900 

.34953 

.27083 

.37143 

11 

12 

.23620 

.30925 

.24759 

.32905 

.2.5920 

.34988 

.27103 

.37180 

12 

13 

.236:39 

.3t)9o7 

.24778 

.32939  1 

.259:39 

.35024 

.27123 

.37218 

13 

14 

.236.i8 

.:309S9 

.24797 

.;32973 

.25959 

.35060 

.27143 

.37255 

14 

15 

.23677 

.31022 

.24816 

.3:3007 

.25978 

.3.5095 

.27163 

.37293 

15 

16 

.2:3696 

.31054 

.24835 

.3:3041 

.2.5998 

.351:31 

.27183 

.37330 

16 

17 

.2:3714 

.31086 

.24854 

.3:3075 

.26017 

.35167 

.272113 

.37368 

17 

18 

.2:37:33 

.31119 

.24874 

..3:31(^ 

.26a37 

.35203 

.27223 

.37406 

IS 

19 

.23752 

.31151 

.24893 

.3:3143 

.26056 

.a52:38 

.27243 

.37443 

19 

20 

.23771 

.31183 

.24912 

.33177 

.26076 

.35274 

.27263 

.37481 

20 

21 

.23790 

.31216 

.24931 

.3:3211 

.20096 

.35.310 

.27283 

.37519 

21 

22 

.2:3.S<i8 

.31248  1 

.24950 

.3:3245 

.26115 

.35346 

.27303 

.37556 

22 

23 

.2:38-27 

.31281  ; 

.24970 

.3:3279 

.26135 

.35382 

.27323 

.37594 

23 

24 

.2:3.^46 

.31313  1 

.24989 

..3:3.314 

.26154 

.35418 

.27343 

.37632 

24 

25 

.2:3S65 

.31:346 

.25008 

..3:3:348 

.26174 

.35454 

.27363 

.37670 

25 

26 

.238,S4 

.31:378  ' 

.25027 

.3:3:382  ; 

.26194 

.35490 

.27383 

.37708 

26 

27 

.2:39<3;3 

.31411 

.25047 

..3:i416  1 

.26213 

.35526 

.274C3 

.37746 

27 

28 

.2:3922 

.3144:3 

.25<:>06 

.3:3451  ' 

.262:3:3 

.35562 

.27423 

.37784 

28 

29 

.2:3941 

.31476  i 

.25085 

.3S485 

.262.53 

.a5598 

.27443 

.37822 

29 

30 

.23959 

.31509 

.25104 

.33519  , 

.26272 

.35634 

.27403 

.37860 

30 

31 

.23978 

.31541 

.25124 

.a3554  ! 

.26292 

.35670 

.2^483 

.37898 

31 

32 

.23997 

.31574 

.25143 

.3:3588 

.26312 

.35707 

.27503 

.37936 

^2 

33 

.24016 

.31607 

.25162 

.33622  . 

.263:31 

.35743 

.27523 

.37974 

33 

S4 

.240:35 

.31640 

.25182 

.a3657  1 

.26.351 

.a5779 

.27543 

.38012 

34 

35 

.240.S4 

.31672  ; 

.25201 

.33691 

.26:371 

..3.5815 

.27563 

.38051 

35 

36 

.24073 

.:31705 

.25220 

.a3726 

.26:390 

.35852 

.27583 

..38089 

36 

37 

.24092 

.317:38  ' 

.25240 

..3:3760 

.26410 

.35888 

.27603 

.38127 

37 

38 

.24111 

.31771 

.25259 

.3:3795 

.2f>4.30 

..a5924 

.27623 

.38165 

38 

39 

.241:30 

.31804 

.2.5278 

..3:38:30 

.26449 

.a5061 

.27643 

.38204 

39 

40 

.24149 

.318:37 

.25297 

.33864 

.26469 

.35997 

.27663 

..38242 

40 

41 

.24108 

.31870 

.25317 

.3.3899  ' 

.25489 

..360.34 

.27&«3 

.38280 

41 

42 

.241'<7 

.31903 

.253:36 

.3:3934 : 

.26509 

.36070 

.27703 

.38319 

42 

43 

.24206 

.319:36 

.253.56 

.3:3968 

.2(3528 

.36107 

.27723 

.38357 

43 

44 

.24225 

.31909 

.25375 

..34003 

.26548 

.36143 

.27743 

.38396 

44 

45 

.24244 

.32002 

.25394 

..340.38 

.26568 

.36180 

.27764 

.384:34 

45 

46 

.24262 

.320:35 

.25414 

.34073 

.26588 

.30217 

.27784 

.38473 

46 

47 

.24281 

.32008 

.254:3:3 

.34108 

.26607 

.362.5:3 

.27804 

.38512 

47 

48 

.24:3«X) 

.32101 

.2.54.52 

.34142 

.26627 

.36290 

.27824 

.38550 

48 

49 

.24320 

.321:34 

.25472 

.ail  77 

.26647 

.36:327 

.27844 

..38.589 

49 

50 

.24:3:39 

.32168 

.25491 

.34212  ! 

.26667 

.36363 

.27864 

.38628 

50 

51 

.24358 

.32201 

.25.511 

.34247 

.26686 

..36400 

.27884 

..a%68 

51 

52 

.24377 

.3-22:34  j 

.25530 

..31282 

.26706 

.36437 

.27905 

.38705 

52 

53 

.24396 

..32267  I 

.25549 

.34:317 

.26726 

..36474 

.27925 

.38744 

53 

54 

.24415 

.32301 

.2.5.569 

.34:3.52 

.26746 

..36.511 

.27945 

.387^3 

54 

55 

.244:34 

.323:34 

.25588 

.34:387 

.26766 

.36.548 

.27965 

.38822 

55 

56 

.244.53 

.32:368  ! 

.25608 

..34423 

.26785 

..36585 

.27985 

.38860 

56 

57 

.24472 

.32401 

.25627 

.34458 

.26805 

.36822 

.28005 

.38899 

57 

58 

.24491 

.324-^ 

.25647 

.34493  i 

.26825 

.36659 

.28026 

.3^9-38 

53 

59 

.24510 

.32468  1 

.2.5666 

.34.528  [ 

.26845 

.36696  . 

.28046 

.38977 

5J 

60 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733 

.28066 

.30C16 

i 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


343 


1 

0 

44° 

45° 

46° 

470 

0 

Vers. 

Exsec. 

.39016 

Vers. 

1 

.29289 

Exsec.  j 
.41421 

Vers. 

1 
Exsec. 

Vers. 

Exsec. 

.28066 

.30.5:34 

.43956 

.31800 

.46628 

1 

.28086 

.39055 

.29310 

.41463  \ 

.30555 

.4^3999 

.31821 

.46674 

1 

2 

.28106 

.39095 

.29330 

.41504  1 

.30576 

.44042 

.31843 

.46719 

2 

3 

.28127 

.39134 

.29351 

.41545 

.30597 

.44086 

.31864 

.46765 

3 

4 

.28147 

.39173 

i  .29372 

.41586 

j  .:30618 

.44129  1 

.318a5 

.46811 

4 

5 

.28167 

.39212 

!  .29392 

.41627 

.306.39 

.44173  \ 

.31907 

.46857 

5 

6 

.28187 

.39251 

1  .29413 

.41669 

.30660 

.44217  t 

.31928 

.46903 

6 

7 

.28208 

.39291 

.294:3:3 

.41710 

.30681 

.44260  1 

.31949 

.46949 

7 

8 

.28228 

.39330 

.29454 

.41752 

.30702 

.44304  i 

.;31971 

.46995 

8 

9 

.28218 

.39369 

.29175 

.41793 

.30723 

.44.347 

.31992 

.47041 

9 

10 

.28268 

.39409 

.29495 

.41835 

.30744 

.44391 

..32013 

.47087 

10 

11 

.28289 

.39448 

.29516 

.41876 

.30765 

.44435 

.32035 

.47134 

11 

VZ 

.28309 

.39487 

.29537 

.41918 

.30786 

.44479 

.32056 

.47180 

12 

13 

.28329 

.39527 

.29557 

.41959 

.30807 

.44523 

.32077 

.47226 

13 

14 

.28350 

.39566 

.29578 

.42001 

.30828 

.44567 

..32099 

.47272 

14 

15 

.28370 

.39606 

.29599 

.42042 

.30849 

.44610 

..32120 

.47319 

15 

l(j 

.28390 

.39646 

.29619 

.42084 

.30870 

.44654 

.32141 

.47365 

16 

17 

.28410 

.;i!)6S5 

.29640 

.42126 

.30891 

.44698 

.;32163 

.47411 

17 

18 

.28431 

.39725 

.29661 

.42168 

..30912 

.447'42 

.32184 

.47458 

18 

19 

.28451 

.39764 

.29681 

.42210 

..30933 

.44787 

.32205 

.47504 

19 

5iO 

.284:1 

.39804 

.29702 

.42251 

.30954 

.44831 

.32227 

.47551 

20 

21 

.28492 

.39844 

.29723 

.42293 

.30975 

.44875 

.32248 

.47598 

21 

22 

.28512 

.39884 

.29743 

.42335 

.30996 

.44919 

..32270 

.47644 

22 

23 

.28532 

.39924 

.29761 

.42377 

.31017 

.44903 

.32291 

.47691 

23 

24 

.28553 

.39963 

.29785 

.42419 

.310.38 

.45007 

..32312 

.477.38 

24 

25 

.28573 

.40003 

.29805 

.42461 

.31059 

.45052 

..32334 

.47784 

25 

26 

.285'J3 

.40043 

.29826 

.42503 

.31080 

.45096 

..32355 

.47831 

26 

2?' 

.28614 

.40083 

.29847 

.42.545 

.31101 

.4,5141 

.32377 

.47878 

27 

28 

.28634 

.40123 

.29808 

.42587 

.31122 

.45185 

.32398 

.47925 

28 

2J 

.28655 

.40163 

.29888 

.42630 

.31143 

.45229 

..32420 

.47'972 

29 

30 

.28675 

.40203 

.29909 

.42672 

.31105 

.45274 

.32441 

.48019 

30 

31 

.28605 

.40243 

.29930 

.42714 

.31186 

.45319 

.32462 

.48066 

31 

32 

.28716 

.40283 

.29951 

.42756 

.31207 

.45363 

.32484 

.48113 

32 

33 

.28736 

.40324 

.29971 

.42799 

.31228 

.45408 

..32505 

.48160 

33 

34 

.28757 

.40364 

.29992 

.42841 

.31249 

.45452 

..32527 

.43207 

34 

35 

.28777 

.40404 

.30013 

.42883 

.31270 

.45497 

..32548 

.48254 

35 

3G 

.28797 

.40444 

.30034 

.42926 

.31291 

.4.5542 

.32570 

.48:301 

36 

37 

.28818 

.40485 

.30054 

.42968 

.31:312 

.45587 

.32591 

.48349 

37 

38 

.28838- 

.40525 

.30075 

.43011 

.31.3:34 

.456:31 

..32613 

.48.396 

38 

39 

.28859 

.40565 

.30096 

.4:3053 

.31.3.55 

.45676 

..32634 

.48443 

39 

40 

.28879 

.40608 

.30117 

.43096 

.31376 

.45721 

.32656 

.48491 

40 

41 

.28900 

.40646 

.30138 

.43139 

.31397 

.45766 

.32677 

.48.5.38 

41 

42 

.28920 

.40687 

.30158 

.4:3181 

..31418 

.4.5811 

.32699 

.48.586 

42 

43 

.28941 

.40727 

.30179 

.4:3224 

.31439 

.4.5856 

.;32720 

.48633 

43 

44 

.28961 

.40768 

,30200 

.43267 

.31461 

.45901 

.32742 

.48()81 

44 

45 

.28981 

.40808 

.30221 

.4.3310 

.31482 

.45946 

.32763 

.48728 

45 

46 

.29002 

.40849 

.30242 

.4:3352 

.31503 

.45992 

.32785 

.48776 

46 

47 

.29022 

.40890 

.30263 

.43:395 

.31524 

.46037 

..32806 

.48824 

47 

48 

.29043 

.40930 

.30283 

.43438 

.31545 

.46082 

.32828 

.48871 

48 

49 

.29063 

.40971 

.30:304 

.4:3-1*^1 

.31.567 

.46127 

..32849 

.48919 

49 

50 

.29084 

.41012 

.30325 

.435:24 

.31588 

.46173 

.32871 

.48967 

50 

51 

.29104 

.41053 

.30^46 

.43567 

.31609 

.46218 

!  ..32893 

.49015 

51 

52 

.29125 

.41093 

.;30:367 

.43610 

.316.39 

.462(i3 

1  ..32914 

.49063 

52 

53 

.29145 

.41134 

.30388 

.4:3653 

.31651 

.4().309 

:  ..329:36 

.49111 

53 

54 

.29166 

.41175 

.30409 

.43696  i 

.31673 

.46:354 

.32957 

.49159 

54 

55 

.29187 

.41216 

.304:30 

.43739 

.31694 

.46400 

.;32979 

.49207 

55 

56 

.29207 

.41257 

.3ai51 

.43783 

.31715 

.46445 

.83001 

.492.55 

56 

57 

.29228 

.41298 

.30471 

.4:3826 

.317.36 

.46491 

.,33022 

.49:303 

57 

58 

.29:248 

.41339 

.30492 

.4:3869 

.31758 

.46.537 

.3.3044 

.49:351 

58 

59 

.29269 

.41380 

.;30513 

.43912 

.:31779 

.46.582 

.3:i065 

.49:399 

59 

60 

.29289 

.41121 

.305^4 

.43956  1 

.31800 

.46628 

.33087 

.49418 

60 

344 


TABLE  XUI.-VERSIXES  AND  EXSECANTS. 


/ 

48» 

49' 

50° 

61» 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

i  Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.33067 

.49448 

.34:394 

.52425 

.35721 

.55572 

.37068 

.58902 

0 

1 

.33109 

.49496 

.:i4416 

.52476 

.35744 

.55626 

.37091 

.58959 

1 

2 

.33130 

.49544 

.344:38 

.52527 

.35766 

.55680 

.37113 

.59016 

2 

3 

.33152 

.49593 

.34460 

.52579 

.35788 

.55734 

.37136 

.59073 

3 

4 

.33173 

.49641 

.34482 

.5263<3 

.35810 

.55789  i 

.3n58 

.59130 

4 

5 

.33195 

.49690 

.34504 

.52681 

.35833 

.5584:3 

.37181 

.59188 

5 

6 

.aS217 

.49738 

.34526 

.52732 

.35855 

.55897 

.37204 

.59245 

6 

7 

.33238 

.497W 

1  .34548 

.52784 

.35S77 

.55951 

.37226 

.59302 

1 

8 

.33260 

.49835  i 

1  .34570 

.52835 

.3590«) 

.56005 

.37249 

.59360 

8 

9 

.33282 

.49884 

1  .34592 

.52886 

.:359->2 

.56060 

.37272 

.59418 

9 

10 

.3:W>3 

.49933 

.34614 

.52938  1 

.35944 

.56114  1 

.37294 

.59475 

10 

11 

.33325 

.49981 

.34636 

.52989 

.35967 

.56169 

.37317 

.59533 

11 

12 

.33:}47 

.500:30 

.34658 

.5:3041 

.35989 

.56223 

.37340 

.59590 

12 

13 

.:B3;368 

.50079  : 

■  .34680 

.53<392  1 

.36011 

.56278 

.37362 

.5964S 

13 

U 

.33390 

.50128 

.34702 

.53144 

.36031 

.56332 

.37385 

.59706 

14 

15 

.3:^412 

.50177 

.34724 

.5:3196 

.36(K6 

.56387 

1  .37408 

.59764 

15 

16 

.334S4 

.50226 

.34746 

.5:3247 

.36078 

.56442 

.37430 

.59822 

16 

IT 

.3*455 

.50275 

.:34768 

.53299 

.36101 

.56497 

.37453 

.59880 

17 

18 

.3:i4rr 

.50:324 

..34790 

.53351 

.36123 

.56551  1 

.37476 

.59938 

18 

19 

.3:i499 

.50:373 

.34812 

.5340:3 

.36146 

.56606 

.37498 

.59996 

19 

20 

.3:^520 

.50422 

.:i4S;34 

.53455 

.36168 

.56661 

.37521 

.60054 

20 

21 

.33.542 

.50471 

.34856 

.53507 

.36190 

.56716 

.37544 

.60112 

21 

22 

.3;55t>4 

.50521 

.34878 

.53559 

.36213 

.56771 

.37567 

.60171 

23 

.3:3586 

.50570 

.34900 

,53611 

.362:35 

.56826 

.37589 

.6>229 

23 

24 

.33607 

.50619 

.34923 

.53663 

.36258 

.56881 

.37612 

.60287 

24 

25 

.3:3629 

.50669  1 

.34945 

.53715 

.36280 

.56937 

.37635 

.60:i46 

25 

26 

.3:3651 

.50718 

.34967 

.53768 

.36302 

.56992 

.37658 

.60404 

26 

27 

.$3073 

.50767 

.34989 

.5:3820  1 

.36:325 

.57047 

.37680 

.60463 

27 

28  1 

.3:3694 

.50617 

.35011 

.53872  1 

.36:347 

.57103 

.37703 

.60521 

28 

29 : 

.3:3716 

..5«D866 

.350:3:3 

.53924  , 

.3<3370 

.571.58 

.37726 

.60580 

29 

30 

.33738 

.50916 

.35055 

.53977 

I 

.36392 

.57213 

.37749 

.60639 

30 

31 

.33760 

.50966 

.35077 

.54029 

.36415 

.57269 

.37m 

.60698 

31 

32 

.3:3782 

.51015 

.35c>99 

.54082 

.364:37 

.57324 

.37794 

.60756 

32 

33  , 

.33803 

.51065 

.3.5122 

.541:34  t 

.36460 

.57380 

.37817 

.60815 

33 

U   ' 

.33825 

.51115 

.35144 

.54187 

.36482 

.57436  1 

.37840 

.60874 

ai 

35 

.33847 

.51165 

.35166 

.54^40 

1  .36504 

.57491 

.37862 

.60933 

35 

36 

.33869 

.51215 

.35188 

.54292 

.36527 

.57547 

.37885 

.60992 

36 

37 

.33891 

.51265  ' 

.35210 

.54345  i 

.36549 

.57603 

.37908 

.61051 

37 

38 

.33912 

-51314  . 

.352:32 

.54393 

.36572 

.57659 

.37931 

•  .61111 

38 

39 

.33934 

.5i:3(>4 

.:35254 

.5«51  , 

,  .36594 

.5. (15 

.37954 

.61170 

39 

40 

.33^6 

.51415 

..35:>77 

.54504  1 

.36617 

.5,..l 

.37976 

.61229 

40 

41 

.33978 

.51465 

.35299 

.54557 

.36639 

.57827 

.37999 

.61288 

41 

42 

.Mm 

.51515 

.35321 

.54610 

.36662 

.57883 

.38022 

.61348 

42 

43 

.34022 

.51565 

.35:^3 

.54663 

.36684 

.57939 

.38045 

.61407 

43 

44 

.34044 

.51615 

.35:365 

.54716 

.36707 

.57995 

.38068 

.61467 

44 

45 

.34065 

.51665 

.35:388 

.54769 

.36729 

.58051 

.38091 

.61526 

45 

46 

.34087 

.51716  I 

.35410 

.54822  ! 

.36752 

.58108 

.38113 

.61.586 

46 

47 

.ail09 

.51766 

.354:32 

.54876 

.36775 

.58164 

.38136 

.61646 

47 

48 

.341:31 

.51817 

.35454 

.54929 

.36797 

.58221 

.38159 

.61705 

48 

49 

.341S3 

.51867 

.35476 

.54982 

.36820 

.58377 

.38182 

.61765 

49 

50 

.34175 

.51918 

.35499 

.55036 

.36842 

.58333 

.38205 

.61825 

50 

51 

.34197 

.51968 

.,35521 

.55089 

.36865 

.58390 

.38228 

.61885 

51 

52 

.34219 

.52019 

.35543 

.5.5143 

.36887 

.58447 

.38251 

.61945 

52 

53 

.34241 

.52«169 

.35565 

.55196 

.36910 

.58503 

.38274 

.62005 

53 

54 

.34262 

.52120 

.a5588 

.55250 

.36932 

.5a560 

.38296 

.62065 

54 

55 

.34284 

.52171 

.35610 

.553a3 

.36955 

.58617 

.38319 

.62125 

55 

56 

.34:306 

.52222 

.a5632 

.55357 

.36978 

.58674 

.38342 

.62185 

56 

57 

.34:328 

.52273 

.35654 

.55411 

.37000 

.58731 

.38365 

.62246 

57 

58 

.34:350 

.52:323 

.a567r 

.55465 

.37i>23 

.58788 

.38388 

.62306 

58 

59 

.34372 

.52:374 

.35699 

.55518 

.37045 

.5*=!845 

.38411 

.62366 

59 

60 

.34394 

.52425 

.35721 

.55572 

.37068 

.58902 

.38434  1 

.62427 

60 

TABLE  Xni.— VERSINES  AN'D  EXSECANTS. 


345 


: 

0 

1  ■ 
52^ 

63° 

54°      i 

55° 

/ 
0 

Vers. 

Exsec.  ' 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

.38434 

.62427 

..39819 

.06164 

.41221 

.701:30 

42&42 

.74345 

1 

.38457 

.6^487 

.39^2 

.66228  1 

.41245 

.70198 

.42«66 

.74417 

1 

2 

.38480 

. 62548 

.39865 

.66292 

.41269  1 

.70267 

.42690 

.74490 

2 

3 

.38503 

.62609 

.39888 

.66357 

.41292  1 

.703a5 

.42714 

.74562 

3 

4 

.38.526 

.62669 

.39911 

.66421  ! 

.41316  ' 

.70403 

.427:38 

.746:35 

4 

5 

.38549 

.627:30 

.399:35 

.66486  ; 

.413:39 

.70472 

.42762 

.74708 

5 

6 

.38571 

.62791 

.39958 

.66550  i 

.41:363 

.70540 

.42785 

.74781 

6 

7 

.38594 

.62852 

.39981 

.66615  ! 

.41:386 

.70609 

.42809 

.74854 

7 

8 

.38617 

.62913 

.40005 

.66679  1 

.41410 

.  7(*677 

.428:3:3 

.74927 

8 

9 

.38640 

.62974 

.40028 

.66744 

.41433 

.70746 

.42857 

.75000 

9 

10 

.38663 

.63035 

.40051 

.66809 

.41457 

.70815 

.42881 

.75073 

10 

11 

.38686 

.63096 

.40074  ' 

.66873 

.41481 

.70884 

.42905 

.75146 

11 

12 

.38709 

.6.3157 

.40098 

.66938 

.41504 

.7095:3 

.42929 

.75219 

12 

13 

.;38r32 

.6.3218 

.40121 

.67003  ! 

.41528 

.71022 

.4295:3 

.75293 

13 

14 

..38755 

.6:3279 

.40144 

.67068 

.41551 

.71091 

.42976 

.75366 

14 

15 

.38778 

.6:3341 

.40168 

.67133  ' 

.41575 

.71160 

.4:3000 

.75440 

15 

IG 

.38801 

.6.3402 

.40191 

.en  99 

.41599 

.71229 

.43024 

.75513 

16 

17 

.38824 

.6:3464 

.40214 

.67264 

.41622 

.71298 

.43048 

.75587 

17 

18 

.38847 

.6:3525 

.402:37 

.67329 

.41646 

.71368 

.43072 

.75661 

18 

19 

.38870 

.63587 

.40261 

.67394 

.41670 

.71437 

.43096 

.75734 

19 

20 

.38893 

.63648 

.40284 

.67460 

.41693 

.71506 

.43120 

.75808 

20 

21 

.38916 

.6.3710 

.40307 

.67525  ' 

.41717 

.71576 

.43144 

.75882 

21 

22 

..38939 

.63772 

.40331 

.67591 

.41740 

.71646 

.43168 

.75956 

22 

23 

-38962 

.63834 

.40354 

.67656  , 

.41764 

.71715 

.43192 

.76031 

23 

24 

.389^5 

.6.3895 

.40378 

.67722 

.41788 

.71785 

.43216 

.76105 

24 

25 

..39009 

.6:3957 

.40401 

.67788 

.41811 

.71855 

.43240 

.76179 

25 

20 

.390.32 

.64019 

.40424 

.6785:3 

.418:35 

.71925 

.43264 

.76253 

26 

27 

.39055 

.64081 

.4044.8 

.67919 

.41859 

.71995 

.43287 

.76328 

27 

28 

.3907^ 

.64144 

.40471 

.67985 

.41882 

.72065 

.4:3311 

.76402 

28 

29 

.39101 

.64206 

.40494 

.68051 

.41906 

.72135 

.43:3:35 

.76477 

29 

30 

.39124 

.61268 

.40518 

.68117 

1  .41930 

.72205  , 

.43359 

.76552 

30 

31 

.39147 

.043.30 

.40541 

.68183 

.4195:3 

.72275 

.43383 

.76626 

31 

32 

.39170 

.64393 

.40565 

.68250 

.41977 

.72346 

.4:3407 

.76701 

32 

33 

.39193 

.64455 

.405S8 

.68316 

.42001 

.72416 

.4:34:31 

.76776 

33 

34 

.39216 

.64518 

.40011 

.68382 

.42024 

.72487 

.43455 

.76851 

34 

35 

.39239 

.64580 

.406.35 

.68449 

.42048 

.72557 

.43479 

.76926 

35 

36 

.39262 

.64643 

.40658 

.68515 

.42072 

.72628 

.43503 

.77001 

36 

37 

.39286 

.64705 

.40682 

.68582 

i  .42096 

.72698 

.43527 

.77077 

37 

38 

.39309 

.64768 

.40705 

.68648 

i  .42119 

.72769 

.43551 

.77152 

38 

39 

.39332 

.648:31 

.40728 

.68715 

1  .42143 

.72840 

.43575 

.77227 

39 

40 

.39355 

.64894 

.40752 

.68782 

i  .42167 

.72911 

.43599 

.77^03 

40 

41 

.39378 

.64957 

.40775 

.68848 

i  .42191 

.72982 

.43623 

.77378 

41 

42 

.39401 

.6.5020 

.40799 

.68915 

'  .42214 

.73053 

.43647 

.77454 

42 

43 

.39424 

.65083 

.40822 

.68982 

.422:38 

.73124 

.43671 

.77530 

43 

44 

.39447 

.65146 

.40846 

.69049 

.42262 

.73195 

.43695 

.77606 

44 

45 

.39471 

.65209 

.40869 

.69116 

.42285 

.73267 

.43720 

.77681 

45 

46 

.39494 

.65272 

.40893 

.69183 

.42309 

.73338 

.43744 

.  1 1 lOi 

46 

47 

.39517 

.65.3.36 

.40916 

.69250 

.42:3:33 

.7:3409 

.43768 

.77833 

47 

48 

.39540 

.65:399 

.40939 

.69318 

.42357 

.73481 

.43792 

.77910 

48 

49 

.39563 

.65462 

.4096:3 

.69385 

.42381 

.73552 

.43816 

.77986 

49 

50 

.39586 

.65526 

1 

.40986 

.69452 

,   .42404 

.73624 

.43840 

.78062 

50 

51 

.39610 

.65589 

.41010 

.69520 

.42423 

.73696 

.43864 

.78138 

51 

52 

.396.33 

.65653 

.410.33 

.69587 

.42452 

.73768 

.43888 

.78215 

52 

5:3 

.39656 

.65717 

.41057 

.69655 

.42476 

.73840 

.43912 

.78291 

53 

54 

.39079 

■  .65780 

.41080 

.69723 

.42499 

.7:3911 

.439:36 

.78368 

54 

55 

.39702 

.65844 

.41104 

.69790 

.42523 

.73983 

.43960 

.78445 

55 

56 

.39726 

.65908 

.41127 

.69858 

.42547 

74056 

.43984 

.78521 

56 

57 

.39749 

.65972 

.41151 

.69926 

.42571 

.74128 

.44008 

.78598 

57 

58 

.39772 

.66036 

.41174 

.69994 

.4'J595 

.74200 

.44032 

.78075 

58 

59 

.39795 

.66100 

.41198 

.70062 

.42619 

.74272 

.440=.7 

.78752 

59 

60 

.39819 

1  .661W 

.41221 

.70130 

1  .42642 

.74:345 

.44081 

.78829 

GO 

346 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

0 

56° 

1             1 
57° 

58° 

59° 

/ 

Vers, 

Exsec. 

Vers. 

Exsec. : 

Vers. 

Exsec. 

1  Vers. 

Exsec. 

.44081 

.78829 

.45536 

.83608 

.47008 

.88708 

.48496 

.94160 

0 

1 

.44105 

.78906 

.45560 

.83690  ! 

.47033 

.88796 

.48521 

.94254 

1 

2 

.44129 

.78984 

.45585 

. 83773 

.47057 

.88884 

.48546 

.94349 

2 

3 

.44153 

.79061 

.45609 

.83855 

1  .47082 

.88972 

.48571 

.94443 

3 

4 

.44177 

.79138 

.456:i4 

.8:3938 

.47107 

.89060 

.48596 

.945:37 

4 

5 

.44201 

.79216 

t  .45658 

.&4020 

.47131 

.89148 

.48621 

.946:32 

5 

6 

.44225 

.79293 

.45683 

.84103 

.47156 

.89237 

.48646 

.94726 

6 

7 

.44250 

.79371 

.45707 

.84186 

1  .47181 

.89325 

.48671 

.94821 

7 

8 

.44274 

.79449 

.45731 

.84269 

.  .47206 

.89414 

.48696 

.94916 

8 

9 

.44298 

.79527 

. 45756 

.843.52 

.472:30 

.89503 

i  .48721 

.95011 

9 

JO 

.44322 

.79604 

.45780 

.&4435  i 

.47255 

.89591 

.48746 

.95106 

10 

11 

.44346 

.79682 

.45805 

.84518 

.47280 

.89680  ; 

1  .48771 

.95201 

11 

12 

.44370 

.79761 

.45829 

.84601 

.47304 

.89769 

.48796 

.9.5296 

12 

13 

.44395 

.79839 

.45854 

.S4685 

.47329 

.89858 

1  .48821 

.9.5392 

13 

14 

.44419 

.79917 

.45878 

.W768 

.47:354 

.89948 

.48846 

.95487 

14 

15 

.44443 

. 79995 

.4.5903 

.84852 

.47379 

.90037 

1  .48871 

.95583 

15 

16 

.44467 

.80074 

.4.5927 

.849:35 

.47403 

.90126 

1  .48896 

.95678 

16 

17 

.44491 

.80152 

.45951 

.85019 

.47428 

.90216 

.48921 

.95774 

17 

18 

.44516 

.80231 

.4.5976 

.8.5103  ■ 

,  .4745:3 

.90305 

\   .48946 

.95870 

18 

19 

.44540 

.80309 

.46000 

.85187 

.47478 

.90:395 

.48971 

.95966 

19 

20 

.44564 

.80388 

.46023 

.85271 

.47502 

.90485 

.48996 

.90062 

20 

21 

.44588 

.80467 

.46049 

.8.5355 

.47.527 

.90575 

.49021 

.96158 

21 

22 

.44612 

.80546 

.46074 

.85439 

'    .47562 

.90665 

.49046 

.96255 

22 

23 

.44637 

.80625 

.46098 

.85523  : 

,  .47577 

.90755 

.49071 

.96:351 

23 

24 

.44661 

.80704  j 

.46123 

.85608 

.47601 

.90845 

i  .49096 

.96448 

24 

25 

.44685 

.80783  1 

.46147 

.85692 

1  .47626 

.90935 

.49121 

.96544 

25 

26 

.44709 

.80862  i 

.46172 

. 85777 

.47651 

.91026 

.49146 

.96641 

26 

27 

.44734 

.80942  ' 

.46196 

.85861  ! 

!  .47676 

.91116 

'  .49171 

.96738 

27 

28 

.447.58 

.81021 

.46221 

.85946  1 

1  .47701 

.91207 

.49196 

.96835 

28 

29 

.44782 

.81101 

.46246 

.b6031 

.47725 

.91297 

.49221 

.96932 

29 

30 

.44806 

.81180 

.46270 

.86116 

.47750 

.91388 

.49246 

.97029 

30 

31 

.44831 

.81260  i 

.46295 

.86201  ' 

.47775 

.91479 

.49271 

.97127 

31 

32 

.44855 

.81340 

.46319 

.86286 

.47800 

.91570 

.49296 

.97224 

:J2 

33 

.44879 

.81419 

.46:344 

.86371 

.47825 

.91661 

.49321 

.97.322 

:33 

34 

.44903 

.81499  1 

.46368 

.86457 

.47849 

.91752 

.49346 

.97420 

34 

35 

.44928 

.81579  ' 

.46393 

.86542 

.47874 

.91844 

.49372 

.97517 

35 

36 

.44952 

.81659 

.46417 

.86627 

.47899 

.91935 

!  .49397 

.97615 

36 

37 

.44976 

.81740 

.46442 

.86713  ' 

.47924 

.92027 

.49422 

.97713 

37 

38 

.45001 

.81820 

.46466 

.86799  i 

.47949 

.92118 

.49447 

.97811 

38 

39 

.45025 

.81900 

.40491 

.86885 

.47974 

.92210 

I  .40472 

.97910 

39 

40 

.45049 

.81981 

.46516 

.86990 

.47998 

.92302 

.49497 

.98008 

40 

41 

.45073 

.82061 

.46.540 

.87056 

.48023 

.92394 

.49522 

.98107 

41 

42 

.45098 

.82142 

.46565 

.87142 

.48048 

.92486 

.49547 

.98205 

42 

43 

.45122 

.82222 

.46589 

.87229  ; 

.48073 

.9-2578 

.49572 

.98304 

43 

44 

.45146 

.82303 

.46614 

.87315 

.48098 

.92670 

1  .49597 

.98403 

44 

45 

.45171 

.82384 

1  .46639 

.87401 

'  .48123 

.92762 

'  .49623 

.98502 

45 

46 

.45195 

.82465 

.46663 

.87488 

.48148 

.92855 

.49648 

.98601 

46 

47 

.45219 

.82546 

.46688 

.87574  , 

.48172 

.92947 

.49673 

.98700 

47 

48 

.45244 

.82627 

.46712 

.87661  I 

.48197 

.9:3040 

.49698 

.98799 

48 

49 

.45268 

.82709 

!  .46737 

.87748 

.48222 

.93133 

1  .49723 

.98899 

49 

50 

.45292 

.82790 

i  .48763 

.87834 

.48247 

.93226  i 

i  .49748 

.98998 

50 

51 

.45317 

.82871 

.46786 

.87921 

.48272 

.93319 

1  .49773 

.99098 

51 

52 

.45341 

.82953 

.46811 

.88008 

.48207 

.93412 

.49799 

.99198 

52 

53 

.45365 

.a3034 

.468:36 

.88095  , 

.48322 

.9.3505  1 

.49824 

.99298 

53 

54 

.45390 

.83116 

.46860 

.88183 

.48347 

.93598  1 

.49849 

.99398 

54 

55 

.45414 

.83198 

.46885 

.88270 

.48372 

.93692  1 

.49874 

.99498 

55 

56 

.45439 

.&3280 

.46909 

.88357  : 

.48:396 

.9:3785  1 

.49899 

.99598 

56 

57 

.45463 

.8.3.362 

1  .469:34 

.88445 

.48421 

.9.3879 

.49924 

.99698 

57 

58 

.45487 

.8.3444 

.46959 

.88532 

.48446 

.93973 

.499.50 

.99799 

58 

59 

.45512 

.83.526 

1  .46983 

.88620 

.48471 

.04066  1 

.49975 

.99899 

59 

60 

.4S5;:6 

.83008 

.   .47008 

.88708  ! 

.48496 

.941  CO  1 

.50000 

i. 00000 

60 

TABLE  XIII.-VEllSINES  AND  EXSECANTS. 


347 


/ 

60» 

Vers. 

Exsec. 

0 

.50000 

1.00000 

1 

.50025 

1.00101 

o 

.50050 

1.00202 

3 

.50076 

1.00303 

4 

.50101 

1.00404 

5 

.50126 

1.00505 

6 

.50151 

1.00607 

i 

.50176 

1.00708 

8 

.50202 

1.00810 

9 

.50227 

1.00912 

10 

.50252 

1.01014 

11 

.50277 

1.01116 

12 

.50303 

1.01218 

13 

.50328 

1.01320 

14 

.50353 

1.01422 

15 

.50378 

1.01525 

16 

.50404 

1.01628 

17 

.504.29 

1.01730 

18 

.5(1454 

1.01833 

19 

.50479 

1.01936 

20 

.50505 

1.02039 

21 

.50530 

1.02143 

22 

.50555 

1  02246 

23 

.50581 

1.02349 

^ 

.50606 

1.02453 

25 

.50631 

1.02557 

26 

.50656 

1.02661 

27 

.50682 

1.02765 

28 

.50707 

1.02869 

29 

..50732 

1.02973 

30 

..50758 

1.03077 

31 

.50783 

1.03182 

32 

.50808 

1.03286 

33 

.50834 

1.03391 

M 

.50859 

1.03496 

35 

.50884 

1.03601 

36 

.50910 

1.03706 

37 

..50935 

1.03811 

38 

.50960 

1.03916 

39 

.50986 

1.04022 

40 

.51011 

1.04128 

41 

.51036 

1.04233 

42 

.51062 

1.04339 

43 

.51087 

1.04445 

44 

.51113 

1.04551 

45 

.51138 

1.046.58 

46 

.51163 

1.047&4 

47 

.51189 

1.04870 

48 

.51214 

1.04977 

49 

.51239 

1.05084 

50 

.51265 

1.05191 

51 

.5*290 

1.05298 

52 

..51316 

1.05405 

53 

.51341 

1.05512 

54 

.51366 

1.0.5619 

55 

.51392 

1.0.5727 

56 

.51417 

1.05835 

57! 

.51443 

1.05942 

58 

.51468 

1.060.50 

59, 

.51494 

1.061.58 

601 

.51519 

1.06267 

61' 


62" 


63« 


Vers.   Exsec 


Vers. 


Exsec 


.51519 
.51W4 
.51570 
.51595 
.51621 
.51646 
.51672 
..51697 
.51723 
.51748 
.51774 

.51799 
.51825 
.51850 
.51876 
.51901 
.51927 
.51952 
..51978 
.52003 
.52029 

.52054 
..52080 
.52105 
.52131 
.52156 
.52182 
.52207 
.52233 
.52259 


.52310 
.52335 
..52361 
.52386 
..52412 
.52438 
.52463 
.52489 
.52514 
.52540 

.52566 
.52591 
.52617 
.52642 
.52668 
.52694 
.52719 
.52745 
.52771 
.52796 

.52822 
.52848 
..52873 
..52899 
.-52924 
.529.50 
.52976 
.53001 
.53027 
.53053 


1.06267 

1  .53053 

1.06375 

.53079 

1.06483 

.53104 

1.06592 

.53130 

1.06701 

.53156 

1.06809 

.53181 

1.06918 

.53207 

1.07027 

.53233 

1.07137 

.53258 

1.07246 

.53284 

1.07356 

.53310 

1.07465 

.53336 

1.07575 

.53361 

1.07685 

.5.3387 

1.07795 

.53413 

1.07905 

.534;^9 

1.08015 

.53464 

1.08126 

.53490 

1.08236 

.53516 

1.08347 

.5.3542 

1.08458 

.53567 

1.08569 

..53593 

1.08680 

.53619 

1.08791 

.53645 

1.08903 

.53670 

1.09014 

.53696 

1.09126 

..53722 

1.09238 

..53748 

1.09350 

.53774 

1.09462 

.53799 

1.09574 

.53825 

1.09686 

.53851 

1.09799 

.53877 

1.09911 

.53903 

1.10024 

.53928 

1.10137 

.53954 

1.10250 

.53980 

1.10363 

.54006 

1.10477 

.54032 

1.10590 

.540.58 

1.10704 

.54083 

1.10817 

..54109 

1.10931 

.54135 

1.11045 

.54161 

1.11159 

.54187 

1.11274 

..54213 

1.11388 

.54238 

1.11503 

..54264 

1.11617  ! 

.54290 

1.117.32 

.54316 

1.11847 

.54343 

1.11963 

.54368 

1.12078 

.54394 

1.12193  1 

..54420 

1.12309  ' 

..54446 

1.12425  1 

.54471 

1.12540  ' 

..'>1497 

1.126.57 

.54523 

1.12773  1 

.54M9 

1. 128^9  . 

.W575 

1.13005  i 

.54601 

1 

.1.3005 
.13122 
.13239 
.13.356 
.13473 
.13590 
.13707 
.13825 
.13942 
.14060 
.14178 

.14296 
.14414 
.145.33 
.14651 
.14770 
.14889 
.15008 
.1.5127 
:15346 
.15366 

.15485 
.1.5605 
.15725 
.15845 
.15965 
.16085 
.16206 
.16326 
.16447 
.16568 

.16689 
.16810 
.16932 
.17053 
.17175 
.17297 
.17419 
.17541 
.17663 
.17786 

.17909 
.18031 
.18154 

.18277 
.18401 
.18524 
.18648 
.18772 
.18895 
.19019 

.19144 
.19268 
.19393 
.19517 
.19642 
.19767 
.19892 
.20018 
.20143 
.20269 


Vers. 


.54601 
.54627 
.54653 
.54679 
.&4705 
.54731 
.54757 
.54782 
.54808 
.54834 
.54800 

..54886 
.54912 
.54938 
.54964 
.54990 
.55016 
.55042 
.55068 
.55094 
.55120 

..55146 
.55172 
.55198 
.55224 
.55250 
.55276 
.55302 
.55328 
.55354 
.55380 

.55406 
.55432 
.55458 
.55484 
.55510 
.555.36 
.55563 
.55589 
.55615 
.55641 

.55667 
.55693 
.55719 
.55745 
.55771 
.55797 
.55823 
.55849 
.55876 
.55902 

.55928 
.559.54 
.55980 
.56006 
.56032 
.56058 
.56084 
.56111 
.56137 
.56163 


Exsec. 


.20269 
.20395 
.20521 
.20647 
.20773 
.20900 
.21026 
.21153 
.21280 
.21407 
.21535 

.21662 
.21790 
.21918 
.22045 
.22174 
.22.302 
.22430 
.22559 
.22688 
.22817 

.22946 
.2.3075 
.23205 
.23334 
.23404 
.23594 
.23724 
.2:5855 
.23985 
.24116 

.24247  .31 
.24378  -32 
.24509  .33 
.24640 
.24772 
.249(3 
.25035 
25167 
.2.5300  139 
.25432  40 

.25565 
.25697 
.25830 
.25963 
.26097 
.26230 
.26364 
.26498 
.26632 
.26766 

.26900  1 51 

.27035  1 52 

.27169  153 

.27304  54 

.27439 

.27574 

.27710 

.27845 

.27981 

.28117 


348 


TABLE  Xm.— VERSINES  AND  EXSECANTS. 


/ 

0 

64°           \ 

65°           ' 

!                        1 
'           66' 

67» 

0 

Vers. 

Exsec. 

Vers. 

Exsec. 

'   Vers. 

1 
1 

Exsec. 

Vers. 

Exsec. 

.56163 

1.28117 

.57738 

l.:36620  ' 

i    .59326 

1.45859 

.60927 

1.55930 

1 

.56189 

1.28253 

.57765 

l.;36768 

.59353 

1.46020 

.60954 

1.56106 

1 

2 

.56215 

1.28390 

.57791 

l.;36916 

.59379 

1.46181 

.60980 

1.56282 

2 

3 

.56241 

1.28526 

.57817 

1.37064 

'   .59406 

1.46342 

.61007 

1.56458 

3 

4 

.56267 

1.28663 

.57844 

1.37212 

.59433 

1.46504 

.61034 

1.566a4 

4 

5 

.56294 

1.28800 

.57870 

1.37361 

.59459 

1.46665 

.61061 

1.56811 

5 

6 

.56320 

1.28937  , 

.57896 

1.37509 

.59486 

1.46827 

.61088 

1.56988 

6 

7 

.56ai6 

1.29074 

1   .57923 

1.37658 

.59512 

1.46989 

.61114 

1.57165 

7 

8 

.56372 

1.29211 

,   .57949 

1.37808 

:    .59539 

1.47152 

.61141 

1.57342 

8 

9 

.56398 

1.29349 

'    .57976 

1.37957 

:    .59566 

1.47314 

.61168 

1.57520 

9 

10 

.56425 

1.29487  i 

.58002 

1.3810?  : 

1   .59592 

1.47477 

.61195 

1.57698 

10 

11 

.56451 

1.29625 

.58028 

1.382.56 

.59619 

1.47640 

.61222 

1.57876 

11 

12 

.56477 

1-29763 

'    .580.55 

1.38406 

1    .59645 

1.47804 

.61248 

1.580.54 

12 

13 

.56503 

1:29901 

.58081 

1.38556 

.59672 

1.47967 

.61275 

1.58233 

13 

14 

.56529 

1.30040 

.58108 

1.38707 

:    .59699 

1.48131 

.61302 

1.58412 

14 

15 

.56555 

1.30179 

.58134 

1.388.57 

1    .59725 

1.48295 

.61:329 

1.58591 

15 

16 

.56582 

1.30318 

.58160 

1.39008 

'   .59752 

1.48459 

.61:356 

1.58771 

16 

17 

.56608 

1.. 30457 

.58187 

1.39159 

.59779 

1.48624 

.61383 

1.58950 

17 

18 

.56634 

1.30596 

.58213 

1.39311 

.59805 

1.48789 

.61409 

1.59130 

18 

19 

.56660 

1.307:35 

.582-iO 

1.39462 

.59832 

1.48954 

.614:36 

1.59:311 

19 

20 

.56687 

1.30875 

.58266 

1.39614 

.59859 

1.49119 

.61463 

1.59491 

20 

21 

.56713 

1.31015 

.58293 

1.39766 

.59885 

1.49284 

.61490 

1.59672 

21 

22 

.56739 

1.31155  '■ 

.58319 

1.39918 

.59912 

1.49450 

.61517 

1.59853 

22 

23 

.56765 

1.31295 

.5^345 

1.40070 

:    .59938 

1.49616 

.61.544 

1.C0035 

23 

24 

.56791 

1.31436 

.5^372 

1.40222 

.59965 

1.49782 

.61570 

1.60217 

24 

25 

.56818 

1.31576 

.58:398 

1.40375 

.59992 

1.49W8  1 

.61597 

1.60399 

25 

26 

.56844 

1.31717 

.58425 

1.40528 

.60018 

1.50115 

.61624 

1.60581 

26 

27 

.56870 

1.31858  . 

.58451 

1.40681 

.60045 

1.50282 

.61651 

1.60763 

27 

28 

.56896 

1.31999  1 

.58478 

1.408:35 

I   .60072 

1.50449 

.61678 

1.60946 

28 

29 

.56923 

1.32140  ' 

.58504 

1.40988 

;    .00098 

1.50617 

.61705 

1.61129 

29 

30 

.56949 

1.32282  ; 

.58531 

1.41142 

1   .60125 

1.50784-  [ 

.61732 

1.61313 

30 

31 

.56975 

1.32424  '■ 

.58557 

1.41296  ' 

'    .60152 

1.50952 

.617.59 

1.61496 

31 

32 

.57001 

1.32566 

.58584 

1.414.50 

.60178 

1.51120 

.61785 

1.61680 

32 

33 

.57028 

1.32708 

.58610 

1.41605 

.60205 

1.. 51289 

.61812 

1.61864 

33 

34 

.57054 

1.32850  1 

.586:37 

1.41760 

.60232 

1.51457 

.618:39 

1.62049 

34 

35 

.57080 

1.32993  1 

.58663 

1.41914  ' 

.60259 

1.51626 

.61866 

1.622.34 

35 

36 

.57106 

i.asi.ss 

.58690 

1.42070 

1    .60285 

1.51795 

.61893 

1.62419 

36 

37 

.571.33 

1.33278 

.58716 

1.42225 

.60312 

1.51965 

.61920 

1.62604 

37 

38 

.57159 

1.. 3.3422 

.58743 

1.42380 

.60339 

1.52134 

.61947 

1.62790 

38 

39 

.57185 

1.3:3565 

.58769 

1.42536  i 

.60365 

1.52304 

.61974 

1.62976 

39 

40 

.57212 

1.33708  1 

.58796 

1.42692  ; 

.60392 

1.5^474 

.62001 

1.63162 

40 

41 

.57238 

1.33852 

.58822 

1.42848  i 

.60419 

1.52645 

.62027 

1.63348 

41 

42 

.57264 

1.3.3996  ' 

.58849 

1.43005 

.60445 

1.52815 

.62054 

1.635:35 

42 

43 

.57291 

1.34140  1 

.58875 

1.43162  : 

.60472 

1.52986 

.62081 

1.63722 

43 

■^4 

.57317 

1.34284 

.58902 

1.43318  ! 

.60499 

1.531.57 

.62108 

1.6:3909 

44 

45 

.57343 

1.34429 

.58928 

1.4.3476 

.60526 

1.5.3.329 

.621:35 

1.64097 

45 

4G 

.57369 

1.34573 

.58955 

1.43633 

.605.52 

1.53500 

.62162 

1.64285 

46 

47 

.57396 

l.a4718 

.58981 

1.43790 

.60579 

1.53672 

.62189 

1.64473 

47 

48 

.57422 

1.34863 

.59008 

1.43948 

.60606 

1.53845 

.62216 

1.64662 

48 

49 

.57448 

1.35009 

.590:i4 

1.44106 

.60633 

1.54017 

.62243 

1.64851 

49 

50 

.57475 

1.35154  i 

.59061 

1.44264 

.60659 

1.54190 

.62270 

1.65040 

50 

51 

.57501 

1.35300  ' 

.59087 

1.44423 

.60686 

1.54363  ! 

.62297 

1.65229 

51 

52 

.57527 

1.35446 

.59114 

1.44582 

.60713 

1.545:36 

.62:324 

1.65419 

52 

53 

.57554 

1.35592  : 

.59140 

1.44741  1 

.60740 

1.54709 

.62351 

1.65609 

53 

54 

.57580 

1.3,5738  : 

.59167 

1.44900 

.60766 

1.54883 

.62378 

1.65799 

54 

55 

.57606 

1.35885 

.59194 

1.45059 

.60793 

1.5505?   i 

.62405 

1.65989 

55 

56 

.57633 

1.36031 

.59220 

1.45219 

.60820 

1.55231 

.62431 

1.66180 

56 

57 

.57659 

1.36178 

.59247 

1.45378 

.60847 

1.55405 

.62458 

1.66:371 

57 

58 

.57685 

1.36325 

.59273 

1.45539  ; 

.60873 

1.55580 

.62485 

1.66.563 

58 

59 

.57712 

1.36473 

.59:300 

1.45699 

.60900 

1.557.55 

.62512 

1.66755 

59 

60 

.57738 

1.36620  1 

.59326 

iAuS:.d 

.60D27 

1.55930  i 

.62539 

1.66947 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


U9 


0 

68» 

69° 

1     70° 

71° 

!  > 
0 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

.63539 

'  1.66947 

.64163 

1.79043 

'  .65798 

1.92380 

.67443 

2.07155 

1 

.62.-)66 

1  1.67139 

.64190 

1,79254 

.65825 

1.92614 

.67471 

2.07415 

1 

o 

.62593 

1.67332 

.64218 

1.79466 

.65853 

1.92849 

.67498 

2.07675 

2 

3 

.62620 

i  1.67525 

.64245 

1.79679 

.65880 

1.93083 

.67526 

2.07930 

3 

4 

.62647 

1.67718 

.64272 

1.79891 

.6.5907 

1.93318 

.67553 

2.08197 

4 

5 

.62674 

1.67911 

.64299 

1.80104 

.65935 

1.93554 

.67581 

2.08459 

5 

G 

.62701 

1.68105 

1  .64326 

1.80318 

.65962 

1.93790 

.67608 

2.08721 

6 

1 

.62728 

1.68299 

1  .64353 

1.80531 

.65989 

1.94026 

.67636 

2.08983 

7 

8 

.62755 

1.68494 

.64381 

1.80746 

.66017 

1.94263 

.67663 

2.09246 

8 

9 

.62782 

1.6S689 

.64408 

1.80960 

'  .66044 

1.94500 

.67691 

2.09510 

9 

10 

.62809 

1.68884 

.04435 

1.81175 

.66071 

1.94737 

.67718 

2.09774 

10 

11 

.62836 

1  69079 

.64462 

1.81390 

.66099 

1.94975 

.67746 

2.10038 

11 

12 

.62863 

1  1.69275 

.64489 

1.81605 

.66126 

1.95213 

.67773 

2.10303 

12 

13 

.62890 

i  1.69471 

:  .64517 

1.81821 

•:   .66154 

1.95452 

.67801 

2.10568 

13 

14 

.62917 

1.69667 

.64544 

1.820.37 

.66181 

1.95691 

.67829 

2.10834 

14 

15 

.62944 

1.69864 

.64571 

1.82254 

.66208 

1.95931 

.67856 

2.11101 

15 

16 

.62971 

1.70061 

.64598 

1.82471 

.662:36 

1.96171 

.67884 

2.11367 

16 

17 

.62998 

1.70258 

.6462.5 

1.82688 

.66263 

1.96411 

.67911 

2.11635 

17 

18 

.63025 

1.704.55 

.646.53 

1.82900 

1  .66290 

1.96652 

.67939 

2.11903 

18 

19 

.630.32 

1.70653 

.64680 

1.83124 

.66318 

1.96893 

.67966 

2.12171 

19 

20 

.63079 

1.70851 

.64707 

1.83342 

j  .66345 

1.97135 

.67994 

2.12440 

20 

21 

.63106 

1.71050 

.64734 

1.83.561 

.66373 

1.97377 

.68021 

2.12709 

21 

22 

.63133 

1.71249 

.64701 

1.83780 

.66400 

1.97619 

.68049 

2.12979 

22 

23 

.63161 

1.71448 

.64789 

1.83999 

.66427 

1.97862 

.68077 

2.13249 

23 

24 

.63188 

1.71647 

.64816 

1.84219 

1  .66455 

1.98106 

.68104 

2.13520 

24 

25 

.63215 

1.71847 

.64843 

1.84439 

[  .66482 

1.98349 

.68132 

2.13791 

25 

26 

.6.3342 

1.72047 

.64870 

1.84659 

.66510 

1.9S594 

.68159 

2.14063 

26 

27 

.63269 

1.72247 

.64898 

1.84880 

.66537 

1.98838 

.68187 

2.14335 

27 

28 

.63296 

1.72448 

.64925 

1.85102 

.66564 

1.99083 

.68214 

2.14608 

28 

29 

.63323 

1.72649 

.64952 

1.85323 

.66592 

1.99329 

.68242 

2.14881 

29 

30 

.63350 

1.72850 

.64979 

1.85545 

.66619 

1 

1.99574 

.68270 

2.15155 

30 

31 

.63377 

1.73052 

.65007 

1.85767 

1  .66647 

1.99821 

.68297 

2.15429 

31 

32 

.6;i404 

1.73254 

.65034 

1.85990 

i  .66674 

2.00067 

.68325 

2.15704 

32 

33 

.634^31 

1.73456 

.65061 

1.86213 

i  .66702 

2.00315 

.68352 

2.15979 

33 

34 

.63458 

1.73659  1 

.05088 

1.86437 

.66729 

2.00562 

.68380 

2.16255 

34 

35 

.63485 

1.73862 

.6.5116 

1.86661 

i  . 66756 

2.00810 

.68408 

2.16531 

35 

36 

.63512 

1.74065  : 

.6.5143 

1.86885 

!  .66784 

2.01059 

.68435 

2.16808 

36 

37 

.63.539 

1.74269 

.65170 

1.87109 

!  .66811 

2.01308 

.68463 

2.17085 

37 

38 

.6.3566 

1.74473  i 

.65197 

1.87334 

.66839 

2.01557 

.68490 

2.17363 

38 

39 

.63594 

1.74677 

.65225  1 

1.87560 

.66866 

2.01807 

.68518 

2.17641 

39 

40 

.63621 

1.74881 

.65252 

1.87785 

.66894 

2.02057 

.68546 

2.17920 

40 

41 

.63648 

1.75086 

.65279 

1.88011 

.66921 

2.02308 

.68573 

2.18199 

41 

42 

.63675 

1.75292 

.65306  ; 

1.88238 

.66949 

2.02559 

.68601 

2.18479 

42 

43 

.63702 

1.75497 

.65334 

1.88465 

.66976 

2.02810 

.68628 

2.18759 

43 

44 

.63729 

1.75703  : 

.65361 

1.88692 

.67003 

2.03062 

.68656 

2.19040 

44 

45 

.63756 

1.75909 

.65388  1 

1.88920 

.67031 

2.03315 

.686&4 

2.19322 

45 

46 

.63783 

1.76116 

.6.5416  ! 

1.89148 

.67058 

2.03568 

.68711 

2.19604 

46 

47 

.6.3810 

1.76323 

.65443 

1.89376 

.67086 

2.03821 

.68739 

2.19886 

47 

48 

.63838  1 

1.76.530 

.6.5470 

1.89605 

.67113 

2.04075 

.68767 

2.20169 

48 

49 

.63865 

1.707.37 

.6.5497 

1.898.34 

.67141 

2.04.329 

.68794 

2.20453 

49 

50 

.63892 

1.76945 

.65.525 

1.90063 

.67168 

2.04584 

.68822 

2.20737 

50 

51 

.63919 

1.771.S4  i 

.65552 

1.90293 

.67196 

2.04839 

,68849 

2.21021 

51 

52 

.63946 

1.77362 

. 65579 

1.90524 

.67223 

2.0.5094 

.68877 

2.21306 

52 

53 

.63973 

1.77571 

.65607 

1.90754 

.67251 

2.05350 

.68905 

2.21592 

53 

54 

.64000 

1  77780 

.6.5634 

1.90986 

. 67278 

2.05607 

.68932 

2.21878  1 

54 

55 

.64027 

1.77990 

.6.5661 

1.91217 

.67306 

2.05864  ! 

.68960 

2.22165 

55 

56 

.64055 

1.78200 

.6.5689 

1.91449 

.67333 

2.06121 

.68988 

2.22452  : 

56 

57 

. 640S2 

1.78410 

.6.5716 

1.91681 

.67361 

2.06379 

.69015 

2.22740  I 

57 

58 

.64109 

1.78621 

.6.5743 

1.91914 

.67388 

2.06637 

.69043 

2.23028 

38 

59 

.641.36 

1.7S832  i 

.6.5771 

1.92147 

.67416 

2.06896 

.69071 

2.23317 

59 

60 

.64163 

1.79043  1 

.65798 

1.92380 

.67443  1 

2.07155  1 

.69098 

2.23607 

50 

350 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


0 

72° 

7 

3° 

74° 

1 

75° 

0 

Vers. 

1 
Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

.69098 

2.23607 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

1 

.69126 

2.2:3897 

.70791 

2.42356 

.72464 

2.63164 

.74146 

2.86790 

4 

2 

.69154 

2.24187 

.70818 

2.426a3 

.72492 

2  63533 

.74174 

2.87211 

2 

3 

.69181 

2.24478 

.70846 

2.43010 

.72520 

2.63903 

1    .74202 

2.87633 

3 

4 

.69209 

2.24770 

.70874 

2.43337 

.72548 

2.64274 

!    .74231 

2.88056 

4 

5 

.69237 

2.2.5062 

!   .70902 

2.43666 

. 72576 

2.&4645 

i    .74259 

2.88479 

5 

6 

69264 

2.25355 

i    .70930 

2.43995 

1   .72604 

2.65018 

.74287 

2.88904 

6 

7 

.69292 

2.25648 

i   .70958 

2.44324 

'   .726:32 

2.65391 

.74315 

2.89.3.30 

7 

8 

.69320 

2.25942 

!   .70985 

2.44655 

.72660 

2.65765 

.74343 

2.89756 

8 

9 

.69:^47 

2  26237 

'   .71013 

2.44986 

.72688 

2.66140 

.74371 

2.90184 

9 

10 

.69375 

2.26531 

j   .71041 

2.45317 

.72716 

2.66515 

.74399 

2.90613 

10 

11 

.69403 

2.26827 

'   .71069 

2.45650 

.72744 

2.66892 

.74427 

2.01042 

11 

12 

.69430 

2.27123 

.71097 

2.45983 

.72772 

2.67269 

.74455 

2.01473 

12 

13 

.69458 

2.27420 

.71125 

2.46316 

.72800 

2.67047  ! 

.74484 

2.01904 

13 

14 

.69486 

2.27717 

.711.53 

2.46651 

.72828 

2.68025 

.74512 

2.02337 

14 

15 

.69514 

2.28015 

.71180 

2.46986 

.72856 

2.65v405 

.74540 

2.92770 

15 

16 

.69541 

2.28313 

.71208 

2.47321 

1    .72884 

2.68785 

.74568 

2.9:3204 

16 

17 

.69569 

2.28612 

.71236 

2.47658 

1   .72912 

2.69167 

.74596 

2.9.3040 

17 

18 

.69597 

2.28912 

.71264 

2.47995 

.72940 

2.09549 

.74624 

2.94076 

18 

19 

.69624 

2.29212 

.71202 

2.4&333 

.72968 

2.^0931 

.74652 

2.94.514 

19 

20 

.69652 

2.29512 

1   .71320 

2.48671 

.72996 

2.70315 

.74680 

2.04952 

20 

21 

.69680 

2.29814 

!   .71^48 

2.49010 

.73024 

2.70700 

'■    .74709 

2.0.5.392 

21 

22 

.69708 

2.30115 

.71375 

2.49350 

.7:3052 

2.71085 

i    .74737 

2.05832 

22 

23 

.697::55 

2.30418 

.71403 

2.4%91 

.73080 

2.71471 

.74765 

2.96274 

23 

24 

.69763 

2.30721 

.71431 

2.50032 

.73108 

2.71858 

.74793 

2.96716 

ai 

25 

.69791 

2.31024 

!    .71459 

2.50374 

.731:36 

2.72^6 

.74821 

2.97160 

25 

26 

.69818 

2.31328 

.71487 

2.50716 

1    .73164 

2.72635  ' 

.74S49 

2.97604 

26 

HI 

.69840 

2.31633 

.71515 

2.51060 

!    .73192 

2.73024 

.74878 

2.98050 

27 

28 

.69874 

2.31939 

.71543 

2.51404 

.73220 

2.7:3414 

.74906 

2.98497 

28 

29 

.60002 

2.32244 

1    .71571 

2.51748 

.73248 

2.73806 

.74934 

2.98044 

29 

30 

.69929 

2.32551 

.71598 

2.52094 

.73276 

2.74198 

.74962 

2.99393 

30 

31 

.69957 

2.32858 

.71626 

2.52440 

'     73304 

2.74591 

.74990 

2.09843 

31 

32 

.69985 

2.33166 

.71654 

2.52787 

.73:332 

2.74984 

1    .75018 

3.00293 

32 

33 

.70013 

2.3;}474 

1   .71682 

2.53134 

1    .73360 

2.75379 

.75047 

3.00745 

.3:3 

ai 

.70040 

2.33783 

•   .71710 

2.5^482 

.7^388 

2.75775 

!    .75075 

3.01198 

34 

35 

.70068 

2.34092 

i    .71738 

2.5;3831 

.7^416 

2.76171 

.75103 

3.01652 

35 

36 

.70096 

2.31403 

.71766 

2.54181 

.73444 

2.76568 

,    .75131 

3.O2107 

36 

37 

.70124 

2.34713 

.71794 

2.54531 

'   .73472 

2.76966  1 

!    .7.5159 

3.02563 

37 

38 

.70151 

2.. 35025 

.71822 

2.54883 

i    .73500 

2.77:365 

,    .75187 

3.0.3O20 

.38 

39 

.70179 

2.. 3.5336 

.71850 

2.55235 

;   .73529 

2.77765  I 

i    .75216 

3.03479 

39 

40 

.70207 

2.35&49 

.71877 

2.55587 

!   .73557 

2.78166 

1 

.75244 

3.03938 

40 

41 

.70235 

2.3.5962 

.71905 

2.. 55940 

!    .73585 

2.78568 

.75272 

3.04398 

41 

42 

.70263 

2.36276 

.71933 

2.56294 

.7:3613 

2.78970  1 

1   .75300 

3.04860 

42 

4-3 

.70290 

2.36590 

'   .71961 

2.56&49 

.73641 

2.79374 

.75328 

3.05.322 

43 

44 

.70318 

2.36905 

!    .71989 

2.57005 

.73669 

2.79778 

.75356 

3.05786 

44 

45 

.70346 

2.37221 

;   .72017 

2.57361 

;    .73697 

2.80183  ; 

.75385 

3.062.51 

45 

46 

.70374 

2.37537 

.72045 

2.. 57718 

.73725 

2.80589  ! 

.75413 

3.06717 

46 

47, 

.70401 

2.37854 

.72073 

2.58076 

.73753 

2..S0996 

.75441 

3.07184 

47 

48 

.70429 

2.38171 

.72101 

2.. 5^34 

.73781 

2.81404 

.75469 

3.07652 

48 

49 

.70457 

2.3*489 

.72129 

2.-58794 

i   .73809 

2.81813 

.75497 

3.08121 

49 

50 

.70485 

2.38808 

.72157 

2.59154 

.73837 

2.82223  j 

. 75526 

3.08591 

50 

51 

.70513 

2.39128 

.72ia5 

2.59514 

.73865 

2.826a3 

.75554 

3.09063 

51 

52 

.70540 

2.39448 

.72213 

2.59876 

.73893 

2.8:3045 

.75582 

3.09535 

52 

53 

.70568 

2.39768 

.72241 

2.60238 

.73921 

2.8:3457 

.75610  1 

3.10009 

53 

54 

.70596  ' 

2.40089 

•    .72269 

2.60601 

;   .73950  ! 

2.83871 

.75639 

3.10484  1 

54 

55 

.70624 

2.40411 

.72296 

2.60965 

1   .73978 

2.84285 

.75667 

3.10960 

55 

56 

.70652 

2.40734 

.72324  i 

2.61330 

.74006 

2.84700 

.7.5695 

3.114:37 

5G 

57; 

.70679 

2.41057 

.72352 

2.61695 

.740*4  ! 

2.85116 

.75723 

3.11915 

57 

58 

.70707  i 

2.41381 

.72380 

2.62061 

.74062  ; 

2.8.5.5:33 

.7.5751  1 

3.12.394 

58, 

^9 

.70735 

2.41705 

.72408 

2.62428 

.74090 

2.8.5951 

.7.5780  i 

3.12875    59 

€0 

.70763 

2.42030  1 

.72436 

2.6279G  1 

.   .74118 

2.86370 

.75808 

3.13357    60, 

TABLE  XIII.-VERSINES  AND  EXSECANTS. 


351 


0 

76° 

1 

'            77° 

78° 

1 

1 

7 

9° 

0 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

1  Vers. 

Exsec. 

.75808 

3.13357 

.77505 

3.44541 

.79209 

3.80973 

1    .80919 

4.24084 

1 

.75836 

3.13839 

.77533 

3.45102 

.79237 

3.81033 

.80948 

4.24870 

1 

2 

.75864 

3.14323 

.77562 

3.45064 

.79266 

3.82294 

.80976 

4.25658 

2 

3 

.75892 

3.14809 

.77590 

3.46228 

.79294 

3.82956 

.81005 

4.26448 

3 

4 

.75921 

3.15295 

.77618, 

3.46793 

.79323 

3.83021 

.81033 

4.27241 

4 

5 

.75949 

3.15782 

.77647 

3.47360 

.79351 

3.84288 

,    .81062 

4.28036 

5 

6 

.75977 

3.10271 

. 77075 

3.47928 

.79:^0 

3.84956 

.81090 

4.28833 

6 

7 

.70005 

3.10761 

.77703 

3.48498 

.79408 

3.85627 

.81119 

4.29034 

7 

8 

.70034 

3.172.52 

.77732 

3.49069 

.79437 

3.86299 

.81148 

4.30430 

8 

9 

.76002 

3.17744 

.7.V00 

3.49642 

.79465 

3.80973 

.81176 

4.31241 

9 

10 

.70090 

3.18238 

,  i  i  too 

3.50216 

.79493 

3.87649 

1    .81205 

4.32049 

10 

11 

.76118 

3.18733 

.77817 

3.. 50791 

.79522 

3.88327 

.81233 

4.32859 

11 

12 

.76147 

3.19228 

.77845 

3.51368 

.79550 

3.89007 

.81262 

4.33671 

12 

13 

.76175 

3.19725 

.77874  ■ 

3.51947 

.79579 

3.89689 

.81290 

4.34486 

13 

14 

.76203 

3.20224 

.77902 

3.. 52527 

.79607 

3.90373 

.81319 

4.35304 

14 

15 

.76231 

3.20723 

.77930 

3.. 53109 

.79636 

3.910.58 

.81348 

4.30124 

15 

10 

.76260 

3.21224 

.77959 

3.53692 

.79064 

3.91746 

.81376 

4.30947 

16 

17 

.76288 

3.21726 

.77987 

3.54277 

.79093 

3.92436 

.81405 

4.37772 

17 

18 

.76316 

3.22229 

.78015 

3.. 54863 

.79721 

3.93128 

.81433 

4.38000 

18 

19 

.76344 

3.22734 

.78044 

3.55451 

.79750 

3.93821 

1   .81462 

4.3943') 

19 

20 

.76373 

3.23239 

.78072 

3.56041 

.79r78 

3.94517 

;    .81491 

4.40263 

20 

21 

.76401 

3.23746 

.78101 

3.56632 

.79807 

3.95215 

.81519 

4.41099 

21 

22 

.76429 

3.21255 

.78129 

3.57224 

.79835 

3.95914 

.81548 

4.41937 

22 

23 

.76458 

3.24764 

.781.57 

3.57819 

.79864 

3.90616 

1   .81576 

4.42778 

23 

24 

.70486 

3.25275 

.78186 

3.58414 

.79892 

3.97320 

.81605 

4.43622 

24 

25 

.76514 

3.2.5787 

.7«214 

3.59012 

.79921 

3.98025 

'   .81633 

4.44468 

25 

26 

.;'(i542 

3.26300 

.7S242 

3.59611 

.79949 

3.98733 

.81602 

4.45317 

26 

27 

.76571 

3.26814 

.78271 

3.60211 

.79978 

3.994J3 

.81691 

4.46169 

27 

28 

.76599 

3.27330 

.78299 

3.60813 

.80006 

4.00155 

.81719 

4.47023 

28 

29 

.76627 

3.27847 

.78328 

3.61417 

.80035 

4.00809 

.81748 

4.47881 

29 

30 

.76655 

3.28366 

.78356 

3.62023 

.80063 

4.01585 

.81776 

4.48740 

30 

31 

.76684 

3.28885 

.78384 

3.62630 

.80092 

4.02303 

.81805 

4.49603 

31 

32 

.76712 

3.29406 

.78413 

3.63238 

.80120 

4.03024 

.81834 

4.. 50408 

32 

33 

.70740 

3.29929 

.78441 

3.63849 

.80149 

4.03746 

1   .81862 

4.51337 

33 

34 

.70769 

3.30452 

.78470 

3.64461 

.80177 

4.04471 

.81891 

4.52208 

34 

35 

.76797 

3.30977 

.78498 

3.65074 

.80206 

4.05197 

.81919 

4.53081 

35 

36 

.76825 

3.31503 

.78526 

3.65090 

.80234 

4.05926 

.81948 

4.53958 

36 

37 

.76854 

3.. 32031 

.7'8555 

3.00307 

.80263 

4.066.57 

.81977 

4.54837 

37 

38 

.7'6882 

3.32500 

.78583 

3.00925 

.80291 

4.07390 

.82005 

4.55720 

38 

39 

.70910 

3.33090 

.78612 

3.67545 

.80320 

4.08125 

.82034 

4.56605 

39 

40 

.70938 

3.33622 

.7'8640 

3.68167 

.80348 

4.08863 

.82063 

4.57493 

40 

41 

.70967 

3.34154 

.78669 

3.68791 

.80377' 

4.09602 

.82091 

4.58383 

41 

42 

.76995 

3.34689 

.78697 

3.69417 

.80405 

4.10344 

;    .82120 

4.59277 

42 

43 

.77023 

3.35224' 

.78725 

3.70044 

.80434 

4.11088 

.82148 

4.60174 

43 

44 

.77052 

3.35761 

.78754 

3.70673 

.80462 

4.11835 

.82177 

4.61073 

44 

45 

.77080 

3.36299 

.78782 

3.71303 

.80491 

4.12583 

.82206 

4.61976 

45 

46 

.77108 

3.36839 

.7'8811 

3.71935 

.80520 

4.13334 

!    .82234 

4.62881 

46 

47 

.77137 

3.37380 

.78839 

3.72569 

.80548 

4.14087 

.82203 

4.63790 

47 

48 

.77165 

3.37923 

.78868 

3.73205 

.80577 

4.14842 

.82292 

4.64701 

48 

49 

.77193 

3.38466 

.78896 

3.73843 

.80605 

4.15.7J9 

.82320 

4.65616 

49 

50 

.77222 

3.39012 

.78924 

3.74482 

.80634 

4.16359 

.82349 

4.66533 

50 

51 

.77250 

3.39558 

.78953 

3.75123 

.80662 

4.17121 

.82377 

4.67454 

51 

52 

.77278 

3.40106 

.78981 

3.75766 

.80691 

4.17886 

.82406 

4.68377 

52 

53 

.77307 

3.40656 

.79010 

3.76411 

.80719 

4.18652 

.82435 

4.69304 

53 

54 

.77335 

3.41206 

.79038 

3.77057 

.80748 

4.19421 

.82463 

4.70234 

54 

55 

.77303 

3.41759 

.79067 

3.77705 

.80770 

4.20193 

.82492 

4.71166 

55 

56 

.7739S 

S.4231i4 

.79095 

3.783.55 

.80805 

4.20900 

.825i.'l 

4.72102 

56  1 

57 

.77420 

3.42867 

.79123 

3.79007 

.80833 

4.21742 

82549 

4.73041 

57 

58 

.77448 

3.43424 

.791.52 

3.79661 

.80862 

4.22.521 

.8257'8 

4.73983 

58 

59 

.77477 

3.4^^982 

.79180 

3.80.316 

.80891 

4.23301 

.82007 

4.74929 

59 

60 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

.82635 

4.75877 

60 

352 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


~0 

80° 

81° 

82° 

83° 

/ 
0 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

.82635 

4.75877 

.84357 

5.39245 

.86083 

6.18.5:30 

.87813 

7.20551 

1 

.82664 

4.76829 

.84:385 

5.40422 

.86112 

6.20020 

.87842 

7.22500 

1 

2 

.82692 

4.77784 

.84414 

5.41602 

.86140 

6.21.517 

.87871 

7.244.57 

2 

3 

.82721 

4.78742 

.84443 

5.42787 

.86169 

6.23019 

.87900 

7.26425 

3 

4 

.82750 

4.79703 

.84471 

5.4:3977 

.86198 

6.24529 

.87929 

7.28402 

4 

5 

.82778 

4.80667 

.84.500 

5.45171 

.86227 

6.20044  I 

.879.57 

7.:3e.388 

5 

6 

.82807 

4.81635 

.84529 

5.46:369 

.86256 

6.27566 

.87986 

7.. 32.384 

6 

7 

.82a36 

4.82606 

.84558 

5.47572 

.86284 

6.29095 

.88015 

7.. 34390 

7 

8 

.82864 

4.a3581  1 

.84586 

5.48779 

.86313 

6.300.30 

.88044 

7.:36405 

8 

9 

.82893 

4.84.558 

.84615 

5.49991 

.86.342 

6.32171 

.88073 

7.. 384:31 

9 

10 

.82922 

4.85539 

.84644 

5.51208 

.86371 

6.. 3:3719 

.88102 

7.40466 

10 

11 

.82950 

4.86524 

.84673 

5.52429 

.86400 

6.. 3.5274 

.88131 

7.42511 

11 

12 

.82979 

4.87511 

.84701 

5 . 53655 

.86428 

6.;3GS:35 

.88160 

7.44566 

12 

13 

.83003 

4.88502 

.847:30 

5.54886 

.864.57 

6.. 38403 

.88188 

7.466:32 

13 

14 

.83036 

4.89497 

.84759 

5.56121 

.86486 

6.. 39978 

.88217 

7.48707 

14 

15 

.83065 

4.90495 

.84788 

5.57361 

.86515 

6. 41. 560 

.88246 

7.. 50793 

15 

16 

.83094 

4.91496 

.84816 

5.58606 

.86544 

6.4.3148 

.88275 

7.52889 

16 

17 

.83122 

4.92501 

.84845 

5.59855 

.86.573 

6.44743 

.88304 

7.. 54996 

17 

18 

.83151 

4.93509 

.84874 

5.61110 

.86601 

6.46:346 

.8a333 

7.57113 

18 

19 

.83180 

4.94521 

.84903 

5.62:369 

.86630 

6 . 4  r955 

.88.362 

7.. 59241 

19 

20 

.83208 

4.95536 

.84931 

5.63633 

.86659 

6.49571 

.88391 

7.61379 

20 

21 

.83237 

4.96555 

.84960 

5.04902 

.866aS 

6.51194 

.88420 

7.6.3.528 

21 

22 

.83266 

4.97577 

.84989 

5.66176 

.86717 

0.. 52825 

.88448 

7.65688 

22 

23 

.8:3294 

4.98603 

.85018 

5.67454 

]   .86746 

6.. 54462 

.88477 

7.67859 

23 

24 

.8a323 

4.99633 

.85046 

5.687:38 

i   .80774 

6.. 56107 

.88506 

7.70^)41 

24 

25 

.83352 

5.00666 

.85075 

5.70027 

'   .86803 

6.577.59 

,88535 

7.722.34 

25 

26 

.a3380 

5.01703 

.85104 

5.71321 

.86a32 

6.. 59418 

.88504 

7.74438 

26 

27 

.8^409 

5.0-3743  1 

.&5i:33 

5.72620 

.86861 

6.61085  1 

.a8593 

7.76653 

27 

28 

.83438 

5.0:3787 

.85162 

5.73924 

.86890 

0.627.59 

.88622 

7.78880 

28 

29 

.a3467 

5.048:34 

.85190 

5.75233 

1   .8()919 

0.64441 

.88651 

7.81118 

29 

30 

.83495 

5.05886 

.85219 

5.76547 

.86947 

0.66130 

.88680 

7.83367 

30 

31 

.a3524 

5.06941 

.85-^ 

5.77866 

'   .86976 

6.67826 

.aS709 

7.85628 

31 

32 

.83553 

5.08000 

.85277 

5.79191 

.87005 

6.69.5:30 

.887:37 

7.87901 

32 

33 

.83581 

5.09062 

.85305 

5.80521 

.870.34 

6.71242 

.88766 

7.90186 

33 

34 

.83610 

5.10129 

.8.5:3:34 

5.81856 

.87063 

6.72962 

.88795 

7.92482 

.34 

35 

.83639 

5.11199  1 

.85:363 

5.83196 

.87092 

6.74689 

.88824 

7.94791 

35 

36 

.83667 

5.12273  1 

.85392 

5.84542 

.87120 

6.76424 

.88853 

7.97111 

36 

37 

.83696 

5.13350 

'    .85420 

5.85893 

.87149 

6.78167 

.88882 

7.99444 

37 

38 

.83725 

5.144:32 

':    .8.5449 

5.87250 

i   .87178 

6.79918 

.88911 

8.01788 

38 

39 

.83754 

5.15517 

1    .85478 

5.88612 

.87207 

6.81677 

.88940 

8.04146 

39 

40 

.83782 

5.16607 

.85507 

5.89979 

.872.36 

6.8.3443 

.88969 

8.06515 

40 

41 

.83811 

5.17700  ' 

.85536 

5.91.3.52 

'   .87265 

6.85218 

.88998 

8.08897 

41 

42 

.83840 

5.18797 

.85564 

5.92731 

1   .87294 

6.87001 

.89027 

8.11292 

42 

43 

.83868 

5.19898  i 

.85593 

5.94115 

!   .87322 

6.88792 

.89055 

8.1.3699 

43 

44 

.83897 

5.21004  ! 

.85622 

5.95505 

.87.351 

6.90592 

.89084 

8.16120 

44 

45 

.8:3926 

5.22113 

.85651 

5.96900 

.87:380 

6.92400 

.89113 

8.18553 

45 

46 

.a39.>4 

5.23226  1 

.85680 

5.98301 

'   .87409 

6.94216 

.89142 

8.20999 

46 

47 

.83983 

5.24:343 

.85708 

5.99708 

.874a8 

6.96040 

.89171 

8. 2^459 

47 

48 

.84012 

5.25464 

.85737 

6.01120 

.87467 

6.97873 

.89200 

8.25931 

48 

49 

.84041 

5.26590 

.&5766 

6.025:38 

.87496 

6.99714 

.89229 

8.28417 

49 

50 

.84069 

5.27719 

.85795 

6.03962 

.87524 

7.01565 

.89258 

8.30917 

50 

51 

.84098 

5.28853 

.85823 

6.05392 

.87.5.53 

7.0.3423 

.89287 

8.. 3.3430 

51 

52 

.84127 

5.29991 

.85852 

6.06828 

,   .87582 

7.05291 

.89316 

8.-35957 

52 

53 

.84155 

5.31133 

.85881 

6.08269 

1   .87611 

7.07167 

.89.345 

8.38497 

53 

54 

.84184 

5.32279 

.85910 

6.09717 

'   .87640 

7.09052 

.89374 

8.41052 

54 

55 

.84213 

5.33429 

.a59:39 

C. 11171 

.87669 

7.10946 

.89403 

8.43620 

55 

56 

.84242 

5.34584 

.85967 

6.126:30 

.87698 

7.12849 

.89431 

8.46203 

^6 

57 

.84270 

5.35743 

.85996 

6.14096 

.87726 

7.14760 

.89460 

8.48800 

57 

58 

.84299 

5.36906 

.86025 

6.1.5568 

.877.55 

7.16681 

.89489 

8.51411 

58 

50 

.84328 

5.:38073 

.86054 

6.17046 

.87784 

7.18612 

.89518 

8.540:37 

59 

60 

.84357 

5.39245 

.86083 

6.18530 

.87813 

7.20551  i 

.89547 

8.5G077 

60 

TABLE  XIII.— VERSINE8  AND  EXSECANTS. 


3  k:  "^ 


84" 


Vers. 


Exsec. 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 

42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60  I 


.89547 
.89576 
.89605 
.896:54 
.89663 
.89692 
.89721 
.89750 
.89779 
.89808 
.89836 

.89865 
.89894 
.89923 
.89952 
.89981 
.90010 
.90039 
.90068 
.90007 
.90126 

.90155 
.90184 
.90213 
.90242 
.90271 
.90300 
.90329 
.90358 
.90386 
.90415 

.90444 
.90473 
.90502 
.90531 
.90560 
.90.589 
.90618 
.90647 
.90676 
.90705 

.90734 
.90763 
.90792 
.90821 
.90850 
.90879 
.90908 
.90937 
.90966 
.00995 

.91024 
.91053 
.91082 
.91111 
.91140 
.91169 
.91197 
.91226 
.91255 
.91284 


8.56677 
8.59332 
8.62002 
8.64687 
8.67387 
8.70103 
8.72833 
8.7.5579 
8.78a41 
8.81119 
8.83912 

8.86722 
8.89547 
8.92389 
8.95248 
8.98123 
9.01015 
9.0392;i 
9.06849 
9.09792 
9.12752 

9.15730 
9.18725 
9.21739 
9.24770 
9.27819 
9.30887 
9.33973 
9.37077 
9.40201 
9.4:3:343 

9.46505 
9.49685 
9.52886 
9.56106 
9.59346 
9.62605 
9.65885 
9.G9186 
9.72507 
9.75849 

9.79212 

9.82596 

9.80001 

9.89428 

9  92877 

9.96348 

9.99841 

10.0:3356 

10.06894 

10.10455 

10.14039 
10.17646 
10.21277 
10.^932 
10.28610 
10.32313 
10.36040 
10.39792 
10.43569 
10.47371 


85= 


Vers, 


.91284 
.91:313 
.91342 
.91371 
.91400 
.91429 
.91458 
.91487 
.91516 
.91545 
.91574 

.91603 
.91632 
.91661 
.91690 
.91719 
.91748 
.91777 
.91806 
.91835 
.91864 

.91893 
.91922 
.91951 
.91980 
.92009 
.920:38 
.92067 
.92096 

.92154 

.92183 
.92212 
.922-11 
.92270 
.92299 
.92328 
.92357 
.92386 
.92415 
.92444 

.92473 
.92502 
.92531 
.92560 
.92589 
.92618 
.92647 
.92676 
.92705 
.92734 

.92763 

.92821 
.92850 
.92879 
.92908 
.92937 
.92966 
.92995 
.93024 


Exsec. 


10.47371 
10.51199 
10.55052 
10.58932 
10.62837 
10.66769 
10.70728 
10.74714 
10.78727 
10.82768 
10.86837 

10.90934 
10.95060 
10.99214 
11.03397 
11.07610 
11.11852 
11.16125 
11.20427 
11.24761 
11.29125 

11.33521 
11.37948 
11.42408 
11.46900 
11.51424 
11.55982 
11.60572 
11.65197 
11.69856 
11.74550 

11.79278 
11.84042 
11.88841 
11.93677 
11.98549 
12.03458 
12.08040 
12.13388 
12.18411 
12.23472 

12.28572 
12.33712 
12.38891 
12.44112 
12.49373 
12.54676 
12.60021 
12.65408 
12.708:38 
12.76312 

12.81829 
12.87391 
12.92999 
12.98651 
13  04350 
13.10096 
13.15889 
13.21730 
13.27620 
13.33559 


w 

/ 

Vers. 

Exsec. 

.93024 

13.33559 

0 

.93053 

13.39547 

1 

.93082 

13.45586 

2 

.93111 

13.51676 

3 

.93140 

13.57817 

4 

.93169 

13.64011 

5 

.93198 

13.70258 

6 

.93227 

13.765.58 

7 

.93257 

13.82913 

8 

.93286 

13.89323 

9 

.93315 

13.95788 

10 

.93344 

14.02310 

11 

.9:3:373 

14.08890 

12 

.93402 

14.15527 

13 

.93431 

14.22223 

14 

.93400 

14.28979 

15 

.93489 

14.35795 

16 

.93518 

14.42672 

17 

.93547 

14.49611 

18 

.93576 

14.56614 

19 

.93605 

14.6367'9 

20 

.93634 

14.70810 

21 

.93663 

14.78005 

22 

.93692 

14.85268 

23 

.9:3721 

14.92597 

24 

.93750 

14.99995 

25 

.93779 

15.07462 

26 

.93808 

15.14999 

27 

.93837 

15.22607 

28 

.93866 

15.30287 

29 

.93895 

15.38041 

30 

.93924 

15.45869 

31 

.93953 

15.53772 

32 

.93982 

15.61751 

33 

.94011 

15.69808 

34 

.94040 

15.77944 

35 

.94069 

15.86159 

36 

.94098 

15.94456 

37 

.94127 

16.02835 

38 

.94156 

16.11297 

39 

.94186 

16.19843 

40 

.94215 

16.28476 

41 

.94244 

16.37196 

42 

.94273 

16.46005 

43 

.94302 

16.54903 

44 

.94331 

16.63893 

45 

.94360 

16.72975 

46 

.94:389 

16.82152 

47 

.9«18 

16.91424 

48 

.94447 

17.00794 

49 

.94476 

17.10262 

50 

.94505 

17.19830 

51 

.94534 

17.29501 

52 

.94563 

17.39274 

53 

.94592 

17.49153 

54 

.94621 

17.59139 

55 

.94650 

17.69233 

56 

.94679 

17.79438 

57 

.94708 

17.89755 

58 

.947:37 

18.00185 

59 

.94766 

18.107:32 

GO 

354 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


/ 

87° 

88° 

89° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.94766 

18.10732 

.96510 

27.65371 

.98255 

56.29869 

0 

1 

.94795 

18.21397 

.96539 

27.89440 

.982^4 

57.26976 

1 

2 

.94825 

18  32182 

.96568 

28.13917 

.98313 

58.27431 

2 

3 

.94854 

18.43088 

.96597 

28.38812 

.98:342 

59.31411 

3 

4 

.94883 

18.54119 

.96626 

28.64137 

.98:371 

60.. 391 05 

4 

5 

.94912 

18.65275 

.96655 

28.89903 

1  .9^00 

61.50715 

5 

6 

.94941 

18.76560 

.96684 

29.16120 

1  .9^429 

62.66460 

6 

7 

.94970 

18.87976 

.96714 

29.42802 

1  .98458 

63.86572 

7 

8 

.94999 

18.99524 

.96743 

29.69960 

.9&487 

65.11304 

8 

9 

.95028 

19.11208 

.96772 

29.97607 

i   .98517 

66.40927 

9 

10 

.95057 

19.23028 

.96801 

30.25758 

.98546 

67 . 75736 

10 

11 

.95086 

19.34989 

.96830 

30.54425 

.98575 

69.16047 

11 

12 

.95115 

19.47093 

.96859 

30.8:3623 

.98604 

70.62285 

12 

13 

.95144 

19.59341 

.96888 

31.1.3366 

:  .98633 

72.14583 

13 

14 

.95173 

19.71737 

.96917 

31.436jl 

i  .98662 

73.7.3586 

14 

15 

.95202 

19.&4283 

.96946 

31.74554 

!  .98691 

75.:39655 

15 

16 

.95231 

19.96982 

.96975 

32.060:30 

i  .98720 

77.1:3274 

16 

17 

.95260 

20.098:38 

.97004 

32.38118 

,  .98749 

78.94968 

17 

18 

.95289 

20.22852 

.97033 

32.708:35 

.98778 

80.85315 

18 

19 

.95318 

20.36027 

.97062 

3:3.04199 

.98807 

82.84947 

19 

20 

.95347 

20.49368 

.97092 

a3. 382:32 

.98836 

84.94561 

20 

21 

.9.5377 

2C. 62876 

.97121 

a3. 72952 

.98866 

87.14924 

21 

22 

.95406 

20.76555 

.97150 

34.08380 

.98895 

89.46886 

22 

23 

.9.5435 

20.90403 

.97179 

.34.44539 

.98924 

91.91.387 

23 

24 

.95464 

21.04440 

.97208 

:34. 814.52  ! 

.98953 

94.49471 

24 

25 

.95493 

21.18653 

.97237 

35.19141 

.98982 

97.22.303 

25 

2(5 

.95522 

21.33050 

.97266 

.35.. 576:33 

'   .99011 

10<').1119 

26 

27 

.95551 

21.47635 

.97295 

,35.96953 

,  .99040 
1   .99069 

103.17.57 

27 

28 

.95580 

21.62413 

.97324 

36.37127 

106.4:311 

28 

2'J 

.95609 

21.77386 

.97353 

36.78185 

.99098 

109.8966 

29 

30 

.956;i8 

21.92559 

.97382 

37.20155 

.99127 

113.5930 

30 

31 

,95667 

22.07935 

.97411 

.37.6.3068 

.r.91.56 

117.5444 

31 

32 

.95696 

22.2:3520 

.97440 

38.06957 

.99186 

121.7780 

.32 

33 

.95725 

22.39316 

.97470 

38.518.55 

.99215 

126.3253 

.33 

34 

.957.54 

22.55:329 

.97499 

38.97797 

.99244 

131.2223 

.3-1 

35 

.95783 

22.71563 

.97528 

39.44820 

'  .99273 

1.36.5111 

.35 

36 

.9.5812 

22.88022 

. 97557 

39.92963 

i   .99302 

142.2406 

36 

37 

.95842 

23.04712 

.97586 

40.42266 

.99.3:31 

148.4684 

.37 

38 

.95871 

23.2163; 

.97615 

40.92772 

.99360 

155.2623 

38 

39 

.95900 

23.. 38802 

.97644 

41.44.52c 

i  .99.389 

162.7033 

39 

40 

.95929 

23.56212 

.97673 

41.97571 

.99418 

170.8883 

40 

41 

.9.5958 

23.73873 

.97702 

42.51961 

.99447 

179.9.350 

41 

42 

.95987 

23.91790 

.97731 

43.07746 

.90476 

189.9868 

42 

43 

.96016 

24.09969 

.97760 

43.04980 

.99505 

201.2212 

43 

44 

.96045 

24.28414 

.97789 

44.23720 

.99535 

213.8600 

44 

45 

.96074 

24.47134 

.97819 

44.84026 

.99564 

228.1839 

45 

4(; 

.96103 

24.66132 

.97^48 

4.5.45963 

.99593 

244.5540 

46 

47 

.96132 

24.85417 

.97877 

46.09596 

.99622 

263.4427 

47 

48 

.96161 

25.04994 

.97906 

46.74997 

.99651 

285.4795 

48 

49 

.96190 

25.24869 

.97935 

47.42241 

.99680 

311.5230 

49 

50 

.96219 

25.45051 

.97964 

48.11406 

.99709 

a42.7752 

50 

51 

.96248 

25.6.5546 

.97993 

48.82.576 

.997.38 

380.9723 

51 

52 

.96277 

25.86360 

.98022 

49.. 55840 

.99767 

428.7187 

52 

53 

.96307 

26.07.503 

.98051 

50.. 31290 

.99796 

490.1070 

53 

54 

.96336 

26.28981 

.98080 

51.09027  i 

.99825 

571.9581 

54 

55 

.96365 

26.. 50804 

.98109 

51.891.56  i 

.99855 

686.5496 

55 

56 

.96394 

26.72978 

.98138 

52.71790 

.99884 

858.4369 

56 

57 

.9&423 

26.95.513 

.98168 

5:3. 57046 

.99913 

1144.916 

57 

58 

.96452 

27.18417 

.98197 

54.4.50.53 

.99942 

1717.874 

58 

59 

.96481 

27.41700 

.98^226 

55.-35946 

.99971 

34:36.747 

59 

60 

.90510 

27.65371 

1  .98255 

56.29869  : 

1.00000 

Infinite  1 

60 

XIV.— TRANSITION-CURVE  COORDINATES.         355 

x  =  l{l-  E) 


0° 


y  =  W 


0° 


10 


10' 
20 
30 
40 
50 

10 
20 

30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 

40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 


Dif. 


0.00097 
194 
291 

388 
485 

0.00582 
679 
776 

873 

970 

10G7 

0.01164 
1260 
1357 
1454 
1.551 
1648 

0.01745 

1842 

1939 

2036 

2133 
oooo 

0.02326 
2423 
2520 
2617 
2714 
2810 

0.02907 
3004 
3101 
3198 
3294 
3391 

0.03488 
3585 
3681 
3778 
3875 
3971 

0.04068 
4165 
4261 
4358 
4455 
4551 


0. 


04648 
4744 
4841 
4937 
5034 
5130 

0.05227 
5323 
5420 
5516 
5612 
5709 
5805 


97 
97 
97 
97 
97 

97 
97 
97 
97 
97 
97 

96 

97 
97 
97 
97 
97 

97 
97 
97 
97 
96 
97 

97 
97 
97 
97 
96 
97 

97 
97 
97 
96 
97 
97 

97 
96 
97 
97 
96 
97 

97 
96 
97 
97 
96 
97 

96 
97 
96 
97 
96 
97 

96 
97 
96 
96 
97 
96 
90 


x=  h\  -E) 


E     Dif. 


0.00000 
0 
1 
1 
2 

0.00003 
4 
5 
7 
8 
10 

0.00012 
14 
17 
19 
2'' 
24 

0.00027 
31 
34 
37 
41 
45 

0.00049 
53 
57 
62 
66 
71 

0.00076 
81 
87 
92 
98 
104 

O.OOIIO 
116 
122 
129 
135 
142 

0.00149 
156 
164 
171 
179 
187 

0.00195 
203 
211 
220 
229 
237 

0.00246 
256 
265 
275 
284 
294 
304 


0 
1 
0 
1 
1 

1 

1 
o 

1 
Q 


4 
3 
3 
4 
4 
4 

4 
4 
5 
4 
5 
5 

5 
6 
5 
6 
6 
6 

6 
6 

i 

6 

7 


I 

8 
8 
8 

8 
8 
9 
9 
8 
9 

10 
9 
10 
9 
10 
10 
10 


<^' 


y  =  lC 


C      Dif. 


10°  10' 

20 
30 
40 
50 


11 


13 


14 


lo 


16 


17 


18 


19 


20 


10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 


10 
20 
30 
40 

50 


10 
20 
30 
40 
50 


10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 

10 
20 
30 
40 
50 


0.05901 
5998 
6094 
0190 
6286 

0.06383 
6479 
6575 
6671 
6767 
6863 

0.06959 
7055 
7151 
7247 
7343 
7439 

0.07535 
7631 
7727 

7823 
7919 
8015 

0.08110 
8206 
8302 
8397 
8493 
8588 


0 


,08684 
8780 
8875 
8970 
9066 
9101 

0.09257 
9352 
9447 
9543 
9638 
9733 

0.09828 
9923 
10018 
10113 
1U208 
10303 


0. 


10398 
10493 
10.588 
10683 
10778 
10873 

.10967 
11062 
11157 
11251 
11346 
11440 
11535 


97 
96 
96 
96 
97 

96 
96 
96 
96 
96 
96 

96 
96 
96 
96 
96 
96 

96 
96 
96 
96 
96 
95 

96 
96 
95 
9o 
95 
96 

96 
95 
95 
96 
95 
96 

95 
95 
96 
95 
95 
95 

95 
95 
95 
95 
95 
95 

95 
95 
95 
95 
95 
94 

95 
95 
94 
95 
94 
95 
18S 


E      Dif. 


0.00314 
325 
335 
346 
357 

0.00368 
379 
391 
402 
414 
426 

0.00438 
450 
462 
475 

488 
502 

0.00514 
527 
540 
554 
567 
581 

0.00595 
610 
624 
639 
653 
668 

0.00683 
698 
714 
729 
745 
761 

0.00777 
793 
810 
826 
843 
860 

0.00877 
894 
911 
929 
947 
964 

0.00982 
1001 
1019 
1038 
1056 
1075 

0.01094 
1113 
1133 
1152 
1172 
1192 
1212 


11 
10 
11 
11 
11 

11 
12 
11 
12 
12 
12 

12 
12 
13 
13 
14 
12 

13 
13 
14 
13 
14 
14 

15 
14 
15 
14 
15 
15 

15 
16 
15 
16 
16 
16 

16 
17 
16 
17 
17 
17 

17 
17 

18 
18 
17 
18 

19 
18 
19 
18 
19 
19 

19 
20 
19 
20 
20 
'.'0 
40 


356 


XIV.  —TRANSITION-CURVE  COORDINATES. 


<i>' 


20°  20' 
40 
21 

20 
40 

22 

20 
40 
23 

20 
40 

24 

20 
40 


26 


28 


29 


.30 


20 

40 

20 
40 

20 
40 

20 

40 

20 
40 


y=iC 


C    Dif. 


X  =  l{\  -  E) 


E     Dif. 


11912 
12101 
12289 
12477 

0.12665 
12853 
13040 
13228 
13415 
13602 

0.13789 
13975 
14162 
14348 
14534 
14720 

0.14905 
15091 
15276 
15461 
15645 
15830 

0.16014 
16198 
16382 
16565 
16749 
16932 
17114 


189 
189 
188 
188 
188 

188 
187 
188 
187 
187 
187 

186 
187 
186 
186 
186 
185 

186 
185 
185 
184 
185 
184 

184 
184 
183 
184 
183 
182 
274 


0 


0 


0 


01252 
1293 
1335 
1377 
1421 

01464 
1509 
1554 
1599 
1646 
1693 

01740 
1789 
1838 
1887 
1937 
1988 

0.02040 
2092 
2144 
2198 
2252 
2307 

0.02362 
2118 
2474 
2531 
2589 
2G48 
2707 


41 
42 
42 
44 
43 

45 
45 
45 
47 
47 
47 

49 
49 
49 
50 
51 
52 

52 
52 
54 
54 
55 
55 

56 
56 
57 
58 
59 
59 
90 


«/>' 


30°  30' 
31 

30 
32 

I        30 

30 
30 
30 

30 
30 
30 

30 


33 

\m 

85 

36 
37 
38 

39 

40 
41 
42 
43 
44 

45 
46 
47 

48 
49 
50 


y  =  lC 


C     Dif. 


0.17388 
17661 
17934 
18206 

18478 

0.18749 
19019 
19288 
19557 
19826 
20094 

0.20361 
20627 
20893 
21158 
21423 
21686 

0.21949 
22212 
22474 
22995 
23513 
24028 
24540 

0.25049 
25554 
26057 
26556 
27052 


273 
273 
272 
272 
271 

210 
269 
269 
269 
268 
267 

266 
266 
265 
265 
263 
263 

263 
262 
521 
518 
515 
512 
509 

505 
503 
499 
496 
492 


X  =  1(\  -  E) 


E       Dif. 


0.02797 
2888 
2981 
3075 
3170 

0.03267 
3365 
3464 
3565 
3667 
8771 

0.03876 
3983 
4090 
4199 
4310 
4422 

0.04535 
4649 
4765 
5001 
5241 
5487 
5739 

0.05995 
6256 
6523 
6794 
7070 
7352 


91 
93 
94 
95 
97 

98 
99 
101 
102 
104 
105 

107 
107 
109 
111 
112 
113 

114 
116 
236 
240 
246 
252 
256 

261 
267 
271 
276 
282 


TABLE 

XV.- 

DEFLECTION-AXGLES  FOR  TRANSITION-CURVES. 

Transit  at  P.T.C.,  n"  =  0. 

Tr.  at  quarter-point,  n" 

=  h 

/i° 

I,° 

(5o° 

)-i-Ao- 

-Bo 

(6.°)  -     •  A^       Br 

(^  for 

Bo 

for^  = 

^1  for  ^  = 

n 

^-1 

A^ 

3 

^\ 

4          3 

n 

.0 

.00 

40    go      go 

10°  12° 

14° 

16° 

4°  8°  12°  14°  16° 

.0 

.00 

.0625 

.05 

.0075 

.0025 

.0775 

.05 

.1 

.03 

.01 

.0975 

.1 

.15 

.0675 

.0225 

.1225 

.15 

.2 

.12 

.04 

.1525 

.2 

.25 

.1875 

.0625 

.1875 

.25 

.3 

.27 

.09 

.2275 

.3 

.3.-) 

.3675 

.1225 

.2725 

.35 

.4 

.48 

.16 

.3225 

.4 

.4.5 

.6075 

.2025 

1 

1 

.3775 

.45 

.5 

.75 

.25 

1 

1 

1 

.4375 

.5 

..55 

9075 

.3025 

1       1 

2 

3 

.5025 

.55 

.6 

1.08 

.36 

1 

1      2 

3 

4 

.5725 

1 

.6 

.65 

1  2675 

.4225 

1 

2      3 

5 

7 

.6475 

1       1       2 

.65 

.  1 

1.47 

.49 

1     1 

3      5 

F- 

i 

11 

.7275 

12      3      4 

.75 

1.6875 

.5625 

1      2 

4      7 

11 

17 

.8125 

13      4      6 

.75 

.8 

1.92 

.64 

1      3 

6     11 

17 

25 

.9025 

14      6      9 

.8 

.85 

2.1675 

.7225 

1     2      4 

9     15 

24 

36 

.9975 

2      6    10    15 

.85 

.9 

2  43 

.81 

1     3      6 

12    21 

34 

51 

1.0975 

3      9     15    22 

.9 

.95 

2  7075 

.9025 

1     4      9 

17    .30 

47 

71 

1 1.2025 

1     4     14    22    33 

.95 

1. 

3. 

1. 

1     5     12 

23    41 

64 

97 

1.3125 

1     6    20    31     47 

1. 

DEFLECTION-ANGLES   FOR  TRANSITION    CURVES.    357 


TABLE  XV.-DEFLECTION  ANGLES  FOR  TRANSITION  CURVES. 

Transit  at  mid-point,  n"  =  \. 

Tr.  at  three-quarter  point 

.71" 

=  1 

n 

<^  for 

Ai 

2 

I  ° 
fiifor4-= 
5           3 

^i 

Bltor'^    - 

71 

6°  10°  14°  16° 

8»  12°  16° 

.0 

.05 

.1 

.15 

.2 

.00 

.0075 

.03 

.0675 

.12 

.25 

.2775 

.31 

.3475 

.39 

1       1 
1 
1 
1 

.5625 
.6025 
.6475 
.6975 
.7525 

1     4     11     17 
1    4     10     15 
1    3      8    12 
1     2      7    10 
2      6      8 

.0 

.05 
.1 
.15 
.2 

25 
.3 
.35 
.4 
.45 

.1875 

.27 

.3675 

.48 

.6075 

.4375 

.49 

.5475 

.61 

.6775 

.8125 

.8775 

.9475 

1  0225 

1.1025 

1      4      6 

1      3      4 

12      3 

1      2 

1       1 

.25 
.3 
.35 
.4 

.45 

.5 
.55 
.6 
.65 

•  i 

.75 

.9075 
1.08 
1.2675 
1.47 

.75 
.8275 
.91 
.9975 
1.09 

1.1875 
1.2775 
1.3725 
1.4725 
1.5775 

1 

.5 
.55 
.6 
.65 

.7 

.75 

.8 
.85 
.9 
.95 
1. 

1.6875 

1.92 

2.1675 

2.43 

2.7075 

3. 

1.1875 

1.29 

1.3975 

1.51 

1.6275 

1.75 

1 

1       1 

1      3 

1     2      5 

1  3      8 

2  6    14 

1.6875 
1.8025 
1.9225 
2.0475 
2.1775 
2.3125 

1       1 

.75 
.8 
.85 
.9 
.95 
1. 

Transit  at 
(S|°)  = 

P.C.i,  n"  =  1. 

T  o 

Transit  at  P.T.C.u  or  P. 
flections  from    tangent 
cular  curve. 

(5c°)  =  Y-^e  +  B 

C.,,  de- 
to  cir- 

i 

n 

.0 

.05 

.1 

,15 

.2 

<}>  for 

1=' 

^1 

B.  for    I   = 

Ac 

B.tor\    - 

n 

.0 

.05 

.1 

.15 

.2 

4°  6°   8°  10°  12°  14°  16° 

4°  6°  8°  10°  12°  14° 

16° 

.00 

.0075 

.03 

.0675 

.12 

1. 

1 .0525 

1.11 

1.1725 

1.24 

1    5    12    23   41    64   97 
1   4    11    20  36  58  86 
1   4     9    18  32  51    76 
1   3     8   16  28  44  66 
13     7    13  23  37  56 

2. 

1 .9475 
1.89 
1.8275 
1.76 

1    5    12   23   41    64 
1    4    11    20   36   58 
14      9    18   32   51 
13      8    16   28   44 
13      7    13   23   37 

97 
86 
76 
66 
56 

.25 

.3 

.33 

.4 

.45 

.1875 

.•J7 

.3675 

.48 

.6075 

1.3125 

1.39 

1.4725 

1.56 

1.6525 

12     6    11   20  31   47 

12     5     9   16  26  39 

2     4     7   13  21   31 

1      3     6   10   16  24 

12     4     8   12  18 

1.6875 

1.61 

1.5275 

1.44 

1.3475 

12      6   11    20   31 

12      5     9   16    26 

2      4     7   13   21 

1      3     6   10    16 

1      2     4     8   12 

47 
39 
31 
24 
18 

.25 

.3 

.35 

.4 

.45 

.5 

.55 

.6 

.65 

.7 

.75 

.9075 
1.08 
1.2675 
1.47 

1.75 

1.8525 

1.96 

2.0725 

2.19 

1      2     3     6     9  14 

12     4     6     9 

113     4     6 

12     3     4 

1     1     2 

1.25 
1.1475 
1.04 
.9275 

.81 

12     3     6     9 

12     4     6 

113     4 

1     2     3 

1     1 

14 
9 
6 
4 
2 

.5 
..55 
.6 
.65 
.  1 

.75 
.8 
.85 
.9 
.95 
ll. 

1.6875 

1.92 

2.1675 

2.43 

2.7075 

l3. 

2.3125 
2.44 
2.5725 
2.71 

2.8.525 
3. 

1     1 

.6875 

.56 

.4275 

.29 

.1475 

.00 

1 

1 

.75 

.8 

.85 

.9 

.95 
1 

358         TABLE  XVI.— TRANSITION   CURVE   TABLE. 


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TABLE  XVI.— TRANSITION   CURVE  TABLE. 


359 


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360 


TABLE  XVI.— TRANSITION  CURVE   TABLE. 


^ 
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1— • 

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O  O  O  O  i- 

TABLE    XVI.— TRANSITION    CURVE   TABLE. 


361 


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St  St  St  ct  ^    cocococo-T    ■TiT'T-rin    minminco    cococccot- 


362 


TABLE   XVI.— TRANSITIOX   CURVE   TABLE. 


ooo    ooooo    o  o  c:  o  o    ooooo    ooooo    ooooo    o  o  o  c  o 
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■r-f     TH^— 11— 11— iC^      C?f7iC.t'7.fCO     COCOCCCO"^     ■TrTj''^Tj<lO     iOOOaCO     CDOOOt' 


CO  TfCOO 

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0  W^  CD  00 
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O  r^i—r-c^a    coTfincoi-    QOOi-icorf    oooocoin    t>oeot-o 


I  cj  (?}  w    w  CO  eo  CO  -rji 


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o  1-1 1-1 1-1 1— T-i    oj  oj  CO  CO  Tf<    Ti<  Tji  in  CO  CO    i>-i>oo0505 


1-1  in  in  0 
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0  in  CO  00  in 

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■^  CO  in  Ci  CO 

in  1—1 1—1  CO  in 
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OS    oot-coinco  ot^TOin  o-^t-ooi  noi^Oi^^o  a t>- -^ -rr •■tj  inTri-it-c* 

00001    OiOiOi  aoi  ciQOGDooi^  {-coinm-«<  coiNt-icjsoo  i~^in-^T?o  cocO'^i— ct; 

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1-11-11-11-11-1  Of  ciQi  i7i  ■:>  cocococoeo  tji-iji-<j>"^-<3<  oinininin  mcoococo 


0000 
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71  ys  ma    coi-cccii-i 


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0000s 

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in  in  in  in  in 

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0000 
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1-1       1-1   r-1  1-1   r^  CJ       Ol  01  C*  OJ  CO 


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Ot  -r  :D  X  O 
kT^  iSl  iC  1^  ^ 


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C  >  -T  CD  V     ^ 

CD  CD  CD  O  i.- 


TABLE   XVI.— TIJANSITION    CURVE   TABLE. 


3G3 


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CO  wM  coco 

fc< 

.-1  CO  in  00 

CO  1-lCO  CJC» 
C*  t-  C*  00  -^ 

0*000  00  00 

OSCOTt-OCO 

00  o  cj  in  oo 

1-  in  o  c>  o 

C»  I-  CO  05  CO 

-T  in  Ci  o  CD 

CO  TH  O  OS  05 

CO  CO  in  o  Ci 

O  TH  CO  CO  C7S 

T-iT-iotQin 

TTio  in  CO  t>. 

OOOi-iO»CO 

T--1   1— 1   1— 1    1— 1 

in  CO  00  05  T-l 

1—1  1—1  1—1  1—1  Oi 

CO  in  t^ coo 

Ci  Ci  Ci  Ci  CO 

CO  in  t-  05  TH 

CO  CO  CO  CO  1^ 

S;  coin  00 

o 

y-i  T-{  1-1  a  so 
o 

i-ilO  o  t~  ■># 

1-1  T-  »j  ?»  CO 

c»  o»  «*  in  00 

Tji  in  CO  t-  00 

ON-^tcooo 

o  Ci  in  00 1-1 

Tj<00i-i»0O 

r}<incO00OS 

1-1  coin  00 1-1 

1^  1— 1 11 1— i  o< 

o*eoco-*-<j' 

Ci  Ci  a*  Ci  CO 

ininocot- 

CO  CO  Tji  1^  in 

oooo  050th 

1— 1   TH 

--0  0  00  05 
iC  CJ  0?  -^ 

CJ  CO  OJ  00  w 

O  00  05  OJ  05 

CO  1-1  {-  o  o 
CO  O  T)<  C\»  c< 

CO  00 1~  CO  in 

TT  05  t~  00  1-1 

CO  CO  -f  CO  I- 

1-  in  o  c?!  in 

oooooco 

Tf  -^CC  CO  tH 

05  m  CO  -^00 
TH  -iji  05  CO  in 

1-1  W  CO 

lO  CO  00  1-1  CO 
1-1  rl 

CO  O  CO  i-  1-1 
1-1  ?«  C<  ©«  CO 

m  C5  -f  05  in 
CO  CO  "^  "^  in 

o  CO  Oi  CO  in 

CO  CO  t-  I-  00 

(MOSCD  Tf  Ci 
05  O  O  1-1  Ci 

TH  1—1  1—t 

ooo  CO  in  -r 
CO  CO  -^  m  CO 

TH   1— I  TH  TH  TH 

Eh 

C5Ci 

00  I- Ot^  1-1 

ooin  T-1COO 

•<*<  I-  C5  o  o 

O5  00Tf  o-^ 

OOOOOQO 

Tf  OS  Ci  CO  CO 

o  o  o  o» 

c;  ci  C5  05  o> 

1-1  CO  lO  I-  OS 

1—1    1—1  1—1  1—1    T—1 

00  cr;  CO  I-  I- 
1-1  CO  lO  I-  C5 

c?  Si  ci  o»  ci 

CD  in  -h  -^  CO 
1-ico  1':  <-  05 
CO  CO  CO  coco 

1-1  o  crsoo  CO 
1— CO  -f  cooo 

TJ1  1^  1^  ■^  ■^ 

•rti  CO  TH  CR  CO 

o  Ci  f  m  I- 
in  in  in  in  in 

Tti  thOs  ooo 

05  TH  Oi  -!■  CO 

in  oco  coco 

oooo 

ooooo 

C»  -J<  O  X/  o 

OOOOO 
C*  -^  CO  00  O 

ooooo 

O*  TTCOQOO 

Oi  rr  CD  CO  O 

OOOOO 
OJ  -^  CD  00  O 

ooooo 

Oi  -^CD  00  O 

•:<  CO  ■<*  o 

t-  00  C5  O  0* 
1-t  1— t 

CO  -^  lO  CO  CO 

05  O  1-1  O?  f 
T-c  at  <?i  CJ  OJ 

in  CD  i~  00  o 

CJ  OJ  T»  Ci  CO 

■—  Ci  CO  IT  CO 
CO  CO  CO  CO  CO 

t-  00  o;  o  Oi 

CO  CO  00  Tfi  TJ< 

oooo 
Tt-  oot  o 

r-1 

ooooo 
ct  -r  "—  00  o 

1— 1  1— 1  1— t  1— 1  0< 

OOOOO 

0<  -r  CO  »  O 

ooooo 
c>  -1-  CD  on  o 

CO  CO  CO  CO  -31 

■<9>  •"iji  -^  I*!  in 

ooooo 

C  -^  O  00  o 

in  in  »n  in  CD 

8?8S8 

CO  O  O  CD  I- 

364 


TABLE   XVI.— TRANSITION   CURVE   TABLE. 


ooooo    ooooo 


OOOO     —  ^ 

r-"      1— t  »— «  *— I  t— <  7>      O?  C?  T*  GVJ 


ooooo 

oi  -^  'o  ao  o 

oieo    oc  ecioco  -^ 


ooooo 

CJ  -^  CC  Xi  o 

■^   "^   "Tjl   TJ-  40 


OOOOO 

(>j  -p  a:  3c  o 
in  If;  iO  lO  13 


ooooo 

C^  ■«■  O  CO  cs 
to  13  ?C  50  l- 


"bi 

05  ^  t-OO 

o  ?»«»o 

00  i-i  -^00  «5 

05  ■;>  o  o  in 

1-  00  00  in  n 

in  C5  X  o  to 
00  in  CO  c)  o 

to  o  00  o  in 

05  C5  X  05  05 

■»}<  to  C»  —  C? 

o  1  CO  in  i- 

CO  rf  in  05  in 
OS  cj  in  00  cj 

l-C  1-1  i-c  (?i 

ci  CO  CO  •*■  o 

in  to  I-  00  05 

CR  O  —  C»  CO 

1—1    1— 1    H   T— 1 

in  --o  i  -  X  OS 

T-1   H   1—1   1—1    11 

825??^^ 

"ii 

?<00  rr  iO 

C5  C5  OlOO  l- 

I-  in  Tp  CO  '^ 

c.  I-  in  T»  C5 

to  CO  05  Tf  o 

ino  Tf  CO  n 

Tf  t^OSO  n 

lO  CO  i-  00  oi 

05  OS  O  Oi  O 

o  I  -:>  CO  rr 
1— <  1— 1 1— ( 1— f  1— t 

00  CC  00  X  I- 

in  tot- 00  C5 

1—1    T— 1    1— «    1— 1    1—* 

t- 1-  to  to  to 

O  "  TJ  CO  -^ 

in  in  Tf>  CO  CO 
in  toi-cc  05 

cj  T-.  o  o  OS 

O  n  li  CC  CO 

CO  CO  CO  CO  CO 

fc, 

1-1  rr  t-  -^ 

O  0»  OS  I-  o 

i-i  00  CO  t-  o 

to  OQO  O  -"S" 

ococc  r>  o 

00  CO  C5  O  Tf 

to  —  in  t-  CO 
o*  o»  (?<  CO  in 

t-  -^  05  CO  -^ 

QO  CJ  CO  OJ  00 

CO  o  in  X  OS 

»n  CO  n  o  o 

*"■ 

i-l(J*  0*  so  TT 

«n  CO  1-  05  o 

1— < 

n  CO  ■*  to  X 
1— ( 1— ( 1—1 1— 1 1—1 

O  ff?  TJi  to  CO 

Oi  ?J  o<  w  w 

ocomooo 

CO  CO  CO  CO  -T 

CO  to  OS 'T?  in 
■f  IT  IT  m  in 

> 

.01     .02 

■^  t-  O  ■<1<  o 
1-ci-iCJ 

i^  in  c»  o  CO 
(?» CO  -^  in  to 

00  c  c»  -t  t- 

CO  0?  in  o  CO 

t>.  n  O  n  to 

c»  00 1  oj  o 

O 

e 

i-i  CI  01  Tf  o 

tocoocoto 
1—"  1— •  1— « 

1—1 1—1 1—1 »—« 

Cf  -?»  CO  CO  'T 

n  Oi  W  IJ  CO 

•^  in  to  to  j^ 

CO  -14"  ■^  in  in 

00  OS  O  n  OJ 

CO  tot- 00  OS 
00 1*  cot- 00 

<» 

n  n  n 

T-l  -I  ,1  -rt  n 

fe 

<35  O  t-  1-1  •-t 

o  i-Hooo  in 

l^  05  0  05  s> 

com  i-iO  CO 

05  in  -^  to  ?> 

05  05  ej  00  00 

OO  -r-  Tjit- 
li  t-  to  CO  CO 

ooooo  CO 
n  CO  I-  CO  CO 

m  CO  CO  CO  in 

in  o  I- 1-  OS 

y^Qi^ 

?o  C5 «-'  m  XI 
t— 1 1— 1 1— » 

<?>  to  —  o  ^ 
cj  c?  '^  CO  ^^ 

to  7  J  a:  in  1* 

■^  »n  in  to  i'- 

o  {-  in  CO  o> 

CC'  00  05  O  1 
n  n 

n  O  05  O  OS 

cj  CC  CO  Tt>  in 

H  11  n  n  n 

OS  O  O  n  5^ 
COQOOSOn 
n  n  n  S<  C^ 

H 

OGO 

I-  lO  C'j  ci  in 

C5  10  to  CO  00 

t-  ■*  o  1 1~ 

C»  t-  Tj<  OS  ?? 

CO  n  1-  n  CJ 

^QOnCJO 

iSSS 

Oi  05  05  GC  00 

■I-I  CO  in  i-  OS 

11  11  11  1—1  n 

I- 1-  o  in  -^ 
•^  CO  in  I-  05 

CO  5?  11  05  i- 

ncoinoQO 

CO  CO  CO  CO  CO 

ineonQOto 
o  TJ  Tf  in  I- 

•^  -"T  -"T  TJI  -«< 

CO  O  to  CO  05 
OS  n  c»  •*  in 
Tf  in  in  in  o 

in  o  to  n  to 
I-  OS  o  c'»  CO 

in  in  to  CO  to 

2s 

■WOOtTO 

ttXNOOTfO 

tOWODTfO 

tCNOOTTO 

to  WOOrfiO 

CO  0*  00  rji  o 

cowQO-^o 

CO  •fl'ooo 

OS  1-1  r>  Tf  CD 

n  1—*  1— (  1—1 

t-  05  O  •??  -^ 
y~  y-i  Ot  a*  at 

in  i~  CO  o  "Tf 
cic*  w  coco 

CO  in  to  CO  o 
coco  coco  -^ 

—  CO  T  to  Xi 

•^  ■*  -T  Tf  Tjl 

OS  1  C»  -f  to 

T»<  in  in  in  in 

CO  X  C'J  n 

o  n  CO  in 

CO  oso  -^co 
L-  en  CO  to  o 

in  ii-<  m  CO 
IT  OS  T  o;  in 

rJiCSX  —  X    0O(r»O5OS?J 
n  t-  -^  G-J  OS     t^  to  -^  CO  CO 

OS  OS  CO  o  o 

oi  cj  CO  -^  in 

co  OS  i-x  ■;> 

to  I-  OS  ^  TT 

S) 

T-inO* 

Ol  C>  CO  CO  TJ1 

in  in  to  i-  i-    X  OS  o  n  o» 

ci?  rrm  to  t-- 

X  c:5o  w  CO 

> 
U 


ososoDX    t-t-mTjioo    'Noxto-^    cjostoeoos    innt^cjt-    (jjtoocoto 


oo  o  o 
wco-<9"in 


OOSC5050S      050S0S0S0S      C505XXX      OOi-l-l-tO 

COCOt-XOS     OiOJCOtJ-      inCOt-XOS     OiCJCOT 

nilniin     nnilnil     Ot  '7^  Ot  Oi  Oi 


tc  to  in  in  -?< 
in  o  I-  X  OS 
CJ  I*  o»  w  w 


-t-  JO  CO  CJ  n 
O  n  0»  CO  Tf 

CO  coco  coco 


&H 

CO  t-  iC  Of 

^cotoo 

t-  OS  O  05  CO 

-^  OS  to  5»  o 

S?Sgg2J 

tOODOSX-^ 

CO  to  o  in  n 

xooto  — 

t—  ifl  "^^  1— 1  1— 1 

CO  'T'  OS  CO  -* 
n  (??  CO  to  OS 

CO  OS  O^  CO  X 

CO  I-  CO  OS  in 

11 

11  nW  CO  •^ 

mo  CO  I-  OS 

Onco  -^  to 

H   H   H   H   Tl 

t- OS  —  CO  in 

T-cnOJOl!?* 

t-  OS  —  CO  in 
cj  wcoeoco 

X  o  CO  m  X 

C13  -rr  IT  "^  "^ 

cj-ittDO    ino»oseDtD 

O  n     nWOJCO'^     CDt-XOffJ     iS'COODii^    t-0'<TOOO»     COiiiOiit- 

,_,,_i    ,_,,i,ioj'?j    (NodcocoTi"    Tjiimntoto 


■cHi-ifficoT   inooonco 

o  nnojojoJOJcoeo'^Trincocot-xxosoiiooii'in 


>J 

CO  TT  too 

to  t-  o  in  o» 
ooos  -r  ^  o» 

—  nCO  to  o 
to  CO  CO  to  CO 

■^  OS  -f  OS  m 
ojTToxo 

s^?;sJJ 

X  —  o  m  I- 
OS  11  in  11  o 

I*  to  r>*'»  tn 

C}  to  CO  OJ  CO 

n  ■?*  -* 

in  t-  o  CO  to 

H  n  irl 

OS  CO  «-  —  CO 
^  C<  M  CO  CO 

n  to  0>  J>  rr 

rfi  TTin  in  to 

o  t-  -^  •"  OS 

I-  t-  X  05  OS 

to  in  CC  •;>  ^ 

O  ^  7>  CO  rr 

O  OS  O  05  05 

in  in  CO  I- X 

OSX     t-OfnX     TOSift-O     nCJuOStO     -"int-Xl-     Tj-OTftOtO     -TO-^tOin 


0  0  05  05      O5O5O5O500 

■*to4.-C5    ncoine-os 


xr-t-toto  in-t<co  —  o  osi-incon  osi^irux  inojxfo 
ncoinc-os  — coini-os  ow-^tox  osucomto  xoncoin 
OJejNCJO^    cocoeococo    "^-ttitt-^    Tjunininm    intotototo 


X  01  CO  o 
oi  ■*  in  I- 


TjiXO>tOO     rfiXCltSO     TQOOJCOO     "^XOJCOO     -"^OOCJCOO     -^QOCJCOO 


xosnOiTfi    intoxo5-H 

1—1  1F1  H       11  1—1  1—1  H  .0} 


oj  CO  m  to  00 

0*0*0*0*0* 


OS  o  o>  CO  in 

O*  CO  CO  CO  CO 


CO »-  OS  o  o* 

CO  CO  CO  ■'T  •n" 


CO  TTtOl^OS 

^J*  '^J'  ^J'  '^Ij'  ^^ 


oooo   ooooo 

TCOOOO      OJ-r-OOOO 
1—1     1—1  1—1  1—1  1—1  o* 


OOOOO 
Oi  TtitOXO 
O*  O*  O*  O*  03 


OOOOO 
o>  -r  to  00  o 

coco  coco  TT 


ooooo 

0»  -T  tOODO 

•^  T»i  ■^  -T  in 


ooooo 

O*  -f  to  GO  O 

in  in  in  in  to 


OOOOO 

o>  -^  to  X  o 
to  to  to  CO  I- 


TABLE   XVT.— TRANSITION    CrRVE   TABLE. 


365 


Ioooo   ooooo 


ooooo 

C?  "*  O  00  o 

Oi  c>  r>t  cj  CO 


ooooo 

(T>  -^  ^  00  O 
CC  CO  CO  CO  "^ 


ooooo 

W  TCOGCO 
■^  -^  Tf  •<«<  o 


OOOOO 

oj  -T  o  00  o 

lO  »0  iO  »o  o 


ooooo 

CJ  T  «0  OD  O 
;0  CO  ;o  ^  (.~ 


00  CO  CO  OC  OS 
■^  T-1  00  lOCO 


'oo   ri ,-1  Ti ni ot   n-^-^ir>io   i-ooooi-'   cjcorfici-i 


lOOi-i'-'ira    Tfccsot^'??    T-IC00505— < 
c;J„^^^    eieoict-o    eocooscooo 

ooosocico 

1-1  t-i  0<  Ci  Oi 


to  •^  «o  I-  c* 
c*  i-OJ  t-  eo 


lO  COQOOi  >- 


ei 
Ml 
O 

o 
O 


V 


V 


oj   oiocoDi-o    »ocoi-icio    eoocowoo    eot-f-cinxi   oiwco-^rji   eoi-iostoco 


00003    0505C500S    0050GOOO   coco  I- {-=5 
^eo^-*    iooi-oooj    o^otn^    ooi-jocs 


50  lO  in  T)<  CO 

O  >—  C  J  CO  TT 

W  Ot  (M  <N  <N 


CO  o»  ^  O  OJ    QO  I-  m  Tt  CO 
incot-oooo    CiO'-'OJeo 

WiMOiWW     (NTOCCeOSO 


-i^oeoin  Ci-+-^0  03  O'^OOtO  eOOOOi-O  «OCOCO{-l-  l-J-t-«3CO  OOi-i^«' 

c5  5:  S  -*  o  00 1-  r- 1--  o  CO  I-  CO  05  I-  50  »n  to  00  i-<  o  o  to  co  i-.  o  o  r-.  ?o  to  oi  -^_  o;  if; 

■    ■    ■_;  cieic6-^ni  t-ooos'-'oi  ■»tcoQOOOj  lot-ocjira  co»-iTf"{-o  eocoocoi- 

ii  c^<n.j^u.i  t.   lAjw-^v-;^  T^,-.i-iCj7t  c>»(?<coeoco  co-^-TTT-in  «nineo»to 


5*^  «oo»ng>o  ^;j5Sooo  co<»05CJO  omoiin^  I-Tfojooo  ooi-i-xo 
,-i  »-(»-•>-< cici  oococoTtm  looi-ooQO  050»-i'?jtj< 

O  <^*''*'^*'  SSSSw  eOT}<Tf<io?o  i-t-oco»-i  (Ncojool-  ooi-'co-^w 

cooscjr-i  loio— <o*i-  (oosirtTfio  i-oDoooi  mt-'rr^i-i  C5i~50coao  oi~ocff»i- 

^O?-00  SjcOODl-O  OOOi-OOCO  CJiOCOTfOO  i-OiO-^l-  iWth  so  00  lO  tOODCO»-0 

cieoid  q6-'-<"*ooco  i-coo6-^'-<  aomcoi—^  QOi-i-i-iT  5S05Pn?2?  »gi>ocoto 

cypjuj  ^^„  oieoeoTTO  jowi-oooo  cro.-';<co  -*mi-Qooi  p^co^j-ic 

O5Q01-  ifiOiODCJCO  00COt--^O5  0Jg?O£iCi  OCOC^Tt"5»  QOOOOCOiO  (NCOCOCOin 

r^friaiai  osoJoooct-  ?oio^co-i-c  OQC'—  coo  oo-^-»-i-co  0O-<*00COt-  >-'•<»<  1--OS* 

SSSS  ?^coini--CT>  T=.coini-o--  ^'>}-3''oco  o^co^so  i-osocjco  »o«di-050 

■n-lCJl.     Wi  ^1;^^^^  OJC<Cfi?J(M  eOCOCOCOCO  CO-^-^-^tJ"  TTTfOlOO  lOOlOOCD 


oooo  ooooo  ooooo  ooooo  ooooo  ooooo  ooooo 


•^tOQOO     0?-*t0Q0O 


CJ  -^  to  00  O 


CJ  ^  50  00  o 
CO  CO  CO  CO  "^ 


CJ  rj<  tOOO  O 
•^  T3<  Tf  TJ»  lO 


CJ  Tl>  ?o  00  o 

in  in  m  m  to 


Ci  ■*  to  00  o 

to  to  to  to  L~- 


U-igJ  TT  to 


■*  OO  l~  ■^  o 
05<W  to  •-■  » 


•^  CO  I-  »n  00 

1-1 1-  CO  o  I- 


to  00  »n  o  gj    o»  to  -f  to  gf 
incow  1-11-1    i-ii-icjcoin 


i-ii^O  oco 
I     05  (N  OGO 


00:< 

(N  in< 


>  m  CO 


T-tTiOJW   cocoTfino    toi-oDOio    i-i«-?cort<jn 


to  I-  Oi  O  1-1 

I  i-<T-(  <r*  oi 


eo  -^  to  t-  x 

(N  CJ  (N  W  Of 


a: 

D 
O 


oo  oo 
g^eo  Tf  in 


05  05  00  00  l- 

to  I*  CO  1-1  o» 

t-  Tf  1-1  00  •* 

o  >n-^  mo 

eo  t-  05  (TJ  ■* 

into  to  to  in 

O  O  Ci  03  03 

in  to  I-  00  oj 

O  O  35  05  00 

o  1-1  <N  c>:  -^ 

cc  00  00  i-J- 
in  to  I-  QC  05 

t~  to  to  i.-:  lO 
O  "  0<  CO  rr 

-^eo  (MC*  1-1 
in  to  I-  00  05 

O*Qi0tQtOt 

Q  05  00  I-  to 
O  O  "  CJ  CO 
CO  CO  CO  CO  CO 

_.j_,4..^    oDto^cog*    i-iooooi--  <-i-toinco  i—csi-'^Q 

SItcooco    aDincoc»<w    coin  J.--;  to  ojOii^-toto  i-oor-.in25 

'i-i   i-iwcortin   to t- 00 d  1-1  «2 nibs's  SJ^i«":Si2i! 

1—11—1  1— ii-Hi— 11— lOJ  oyg'OJg^co 


in  C7S  o»  Tf  ■»♦< 
in  i-iOJt-tC 

■^  I-  C35  i?nn 
eoeoco  ■^•<9< 


eooini 

to  t- 00  I 


JO 


00  »-i  T»i  00 1-1 

•"fliinin  in  to 


Of 

o 


in 00 coo  in 
1-1 1-1 1» 


CO  Tf  t^-^  t- 
eo  -*  in  I-  00 


»-icoinoOi-«    TTOOWtOi-i 
i-ii-ii-li-io*    ©ieicoeoTf 


«O<>J00tJ<t-i 

I*  in  in  to  t- 


00  to  Tf  CO  ©I 

l-^  00  050  1-1 


1-1  f-i  01  CO  -^  to 

o 


00  O  CO  I-  o 

1— 1  T-(  »-<  C* 


©jcoso-^in   intot-oooj 


Oi-iwcoin    toi-Oi-i-io? 

,_i-(i-i,-i,-i     I—,-,,—  (jfcj 


s» 


TCOinco    cjc-j-ti-o    coint-cot- 

OOOOCOW      inTJCOOOOO     ^Q005-^CO 


1-1  CO  in 


I-  oco  oo 

1— 1  f-1 1-1  ffl 


irj  05  -r  O  O 

©J  gj  CO  -^  TP 


in  o  5*  o  CO 

to  PS  CO  i-  ■^ 


ir>  03  to  CO  ■« 
in  in  to  I-  oo 


i-i-^OCO-^     l-i-i-ftOtO 

m  o;  I-  cC'  g?    05  o  CO  Ci  00 


CO  to  m  o:  00 

O  ■»  »-i  O  TJ 


05<-toinin    ■^icmicto    ocosi— icom 
00  o3  o  1— 1  2»    CO  "^  m  to  I—   00  05 1— 1 1?»  r? 

1-^1— II— I     1—11—11—11—11—1     1— '1— iC^C?o 


tocootoo    •«i'tot-tOT>>    0'*t-t-in    i-imtOTt-o 


050500  tocootoo  •«i'tot-tOT>>  0'*t-t-in  i-imtOTt-o  -rr  -^  'it  t^  a> 

0050505  0505050000  I—  tOiT^CO  SJOOOtO-^  0J05tCC0O  tOgfOOeOQO 

■^-Si-os  T^coin?-c:5  i-.coint-05  i-co-T<toco  o^comt-  oooi-eo-^ 

^"  i-iiJii-ii-i-i  OJgjiJiJTJ  cococococo  Tjirf^TT-v  T^inininin 


oo  rt  to  iC  1-1 

CO  00  o>  to  d 
to  i~  OS  o  c* 
in  ic  in  to  to 


t0-<*<i?»0     ODtDT}>©»0     00t0T>>(J?O     OOOTtOO     QOtOrfOJO     QOtO'^OJO     OOtO'^WO 

ih  in 


ccint>03   oofi^tooo   05--coin{- 

1-11-1  1—1  1-1  1-1     1—1  ©J  i^  o*  c* 


00  o  cf  ■»*■  to    I-  05 1—  CO  lO 

©»  CO  CO  CO  CO     CO  CO  "^  !*■  T 


tOQD< 
rji  ij>  1 


IC  <  -  05  ^-  CO 

m  m  m  t£  to 


OOOQ     ooooo 

TptooDO    o»irocoo 
1—1    1—1 1—1 1—1 1^  g* 


ooooo  ooooo  ooooo  ooooo 

OJTtOOOO     *»'*O0DO     OJ-PtOODO      WTPtOODO 

©lOJCfOJCo    cococoeois"    ■^■*i*"Ti'in    ininmmto 


oo  ooo 

o>  "  »  «  o 

to  CO  to  to  {- 


r^GG 


TABLE   XVI.— TRAXSITIOX   CUKVE   TABLE. 


- 

- 

|35?| 

—  1-  —  ^  5<< 

CJ  3>J  ??  T>  ro 

•?5  ^  3  X  S 

CO  «  r3  CO  .n" 

i^SxS 

rr  —  -r  r'  1.-3 

1.0  lT.  lt:  0  to 

00000 
rj  7T  to  X  p 

> 

o 

T 

"si 

Tf  —  --C  l~ 

—  ro  LT  X 

■?<  i^  ?<  i~  ■«»• 

tr;  s»  o  ■^  cs 

T-i  o  I-  « in 

CS  iQ  £-  W  rr 

0  0  1.0  -^  .0 
•rl-O  ^X 

—  0  — TTO 
CO  X  CO  X  -^ 

C50  CO  to  ■-- 

Ci  to  ■7J  X  in 

^  T-  Jj  ?*  eo 

TT  TJ.  O  »  l- 

X  c;  e  —  CO 

~r  '~.  i-  X  c» 

1—1  0)  T  in  £~ 

at  ^7}  a  OJ 

f,^u^^ 

« 

C5  X  J-iS-J- 

?JOi--!-  = 

O  ?>  i2  O  TT 

I,  ~  1-1  ri  « 

0  0;  to  coo 

rr  t~  0  7J  CO 

iS  2  i-  X  C5 

c:  s;  X  X  X 

O  —  1?  :o  ■* 

L^  £-  -.O  13  1.': 
L-  O  i-  X  S5 

0  1-1  OJ  CO  -* 
C>7*  NO»S* 

0  X  £-  to  TJ. 

in  in  to  £^  X 

CI  W  C<  5*  5* 

co^o  X  to 

C3  0  •"  1—  7* 

o>  CO  en  CO  00 

&H 

»o  -^  •v  ;s  Ci 

?J  M  Tl<  O  -O 

X  t-{-  — X 

r?  c:  O  L.-  r- 

00  C5  .-^  W  in 

X  —  L-  CO  ?? 

m  X  —  ^  7j 

i-  ci  I?  -T  l-^ 

1-  -^  7*  ;j  7i 

.^  t-  OJ  I-  CO 

ci  o>  m  X  « 

C?  CO  CO  CO  "^ 

O?  0  0  X  £- 

OJ  CO  0  CO  CO 

d  x'  71  d  d 

•^  -3-  uo  m  «o 

in  CO  X  CO  X 
1-1 0  c;  0  1— 

d  f-  0  d  d 

• 
O 

30  Tj>  —  ^  -> 

;i  ■'S'  o  X  — 

X  t-  CJ  m  -^ 
O  I-  CS  TJ  O 

lO  X  r?  =;  o 
1-1  —  7»  ?»  ^?' 

O  CO  I-  5»  £- 

1-  IJ  7J  ec  CO 

CO  c;  -.o  -r  JO 
■^  -r  lO  —  £- 

Ci  0  0>  CO  0 

0?  ?J  i?  --  to 

X  d  d  —^  0? 

CO  X  0  OJ  Tfi 

CiCOX-^^ 

CO  1.0  CO  xd 

1— «  l-H    t— t  1— •    7? 

CO  X  0  7?  1* 

—  .— '  1— 1  1— 1 

—  —  0?  o»  0? 

d  7?  CO  CO  CO 

;a 

■«  o  r?  CM~ 

O  JO  I-  ?J  i- 

^  ^  —  ?» w 

C*^  *  '-^  c^  ^^ 

7>  Of  o  CO  -;> 

OJ  X  X  CO  ?? 

ci  e-  o  ^  o 

;o  i>"  X  C5  o 

0  I 0  0> 

IC  —  0>  to  CO 

CO  i-  X  d  -^ 

—  0»  CO  -rr  to 

COi^CO0l« 

d  d  x  — ^  -«' 

0  to  0  —  t- 
I-  7?OC:  CI 

£-  ■—  m  X  7) 

CO  m  to  £-  0 

7J  7>  7*  7>  7» 

cix-o 

c?  X  ;j  o  o 
c;  X  X  I-  » 

•^  ^  ~r  ifi  -^ 
O  -r^  •:>=:'  X 

—  TCi-l-X 

c:  0?  -^  :o  X 

O  CO  -6  -o  0) 
o  o>  -^  .o  i- 
co  CO  CO  CO  ec 

CO  T-i  ^  t-  05 

X  -r  d  CO  £-^ 

OC  0  —  CO  T»< 

CO  d  £-  —  C5 

— '  ^  £-  0  — ' 

to  £-  X  0  — 

-r  •^  T  in  in 

CO  7?  to  -T  X 

CO  -f  -r  -r  CO 
7J  CO  -fin  to 

»n  in  in  in  ».n 

2. 

XTJOO 

'^  X  C'J  :C  O 

rr  u;  r: r 

*-^  1— 1   r^  T*   T^ 

f  00  ?» o  o 

?c  X  —  CO  :o 

7>  T»  M  W  M 

X  d  CO  lO  X 

eo  TT  .3.  —  Tj. 

-r  X  7J  CO  0 

0  oj  01-  0 

0  iiO  0  uo  to 

."S-  X  ?$  to  0 
o>  -r  £-  d  7> 

CO  CO  to  CO  £- 

t  X  7>  to  0 

-r  to  SJ  ^  t 
I-  £-  £-  X  06 

> 

-J 

o 
-^» 

5s 

i.'^  to  «  ?-  o 

■?-H   O   O   lO    ^— 

1-1  TO  O  ~>  O 
X  L-  CO  .—  o 

CO  7?  LO  CO  <- 

OS  Ci  S-.  0  -- 

'T  to  OJ  0>  to 

CO.OX-.-T 

-1"  uO  0  t-  £- 
X  7>  i-  "  CO 

c.  -r  0  s;  s: 

^  i-CC  X  -T 

»-n-H  ^  ;» « 

CO  TT  L'J  O  t- 

i-  X  ci  —  0? 

CO  -f  0 1-  X 

0:  ■—  7J  -r  0 

1-  7J  7*  7J  7J 

£-  X  0  —  CO 

7J  7J  CO  CO  r? 

H 

o 

7>  ■n  T  T 

c;  X  i~  -.i  L- 

ci  jrj  :r;  ~.  « 
«--:  :o  I-  X  Cl 

CO  1— *  Ci  O  CO 

C5  si  X  X  X 

O  ^  "J  CO  -T 

O'-O  7J{-— . 

X  i~  t-  d  d 
i~  ;s  i-  X  C5 

LO  cr;  —  CO  0 
i-o  -r  -r  CO  oj 

0  —  OJ  CO  -T 

0  Ot  0»  0*  0> 

to  to  un  -r  7» 
In  3  Si-x 

OJ  7>  7J  7J  0» 

c;  m  —  to  S-. 
1.0  -^  CO  -^  3: 

C5  0  —  7'  7? 
7>  CO  CO  CO  CO 

&H 

ox  ->o 
?*  ci  o  -o 

^—1  »-< 

o  r?  X  o  {- 

rr  —  o  — M 

<rj « .«•  ir>  o 

OiCO  —  IJ-TJ 
tS  —  t-  -T  ?» 

I-  C5  O  1?  TT 

1.0  —  X  -^  1.0 

1-  0?  CO  to  0 

to  x"o  -:>  0 

1—  ^  T*  C»  0> 

cox  -OlftO 

in  —  cs  I- CO 

{-  0  0>  d  X 

CJ  CO  CO  CO  CO 

fCS^XCO 
<-  X  -^  -r  C3 

^  -r  x'  d  •.+ 

■»  TT  TTin  0 

^0  X  3=!^ 

X  7?  d  d  d 

m  CO  to  to  {- 

■•—1 

o 

l-  C»  X  t~  I- 

Oiftcoirt  O 
»--:  ;i>  X  o  CO 

«  c;  CO  t-^ 

CO  cj  X  «n  i» 

OXt-£-t- 

X  0  '7?  10  0 

c<  CO  •*  ;o  C5 

'NlCC5-1>CS 
^  ^  -.  CJ  T? 

■~"7*  7>C0 

Tji  TT  m  » I- 

CO  TJ. -<9.  0  CO 
X  C5  0  1-  7* 

t-  t-  OD  CS  0 

Tji  m  I-  X  0 

1-  CO  -f  m  £- 

7>  1*  to  X  0 

1— »  1— »  1—1 

1.^  1-^  1— 1  1--  7> 

7?  7J  7»  71  ro 

■?»  o  n  X 

X  X  {-  ;o  r? 

•^  —  -:»  u-  ro 

gi^::^ 

X  LO  i.O  m  lO 
i~  I-  .-  Oi  '-i 

0  OJ  lO  CO  iO 
I-  I-  0  I-  I- 

Oto  — -r-r 
1-1 1-  I-  c;  Tj. 

Oi  I-  {>  £-  to 

^  t-i  CO  I-  CO 

"  s*  ■«»•  o 

C5  i»  -o  o  o 

'-'  i-i  5J  CM 

O  =C  ?'  Ci  -o 

CO  CO  rr  -T  O 

co -H ox  X 

CO  i-XX  S5 

£-i-  X  X  s; 

0  1-1  7J  CO  -r 

0  i'-  X  ?.  p 

7>  in  X  — .  m 

7>  CO  -T  to  i- 
7>7J7>7'  7f 

i7 

CiXO 

TT  O  1"  SI  '-' 

^  o  o  o  ■:> 

•^X  7»  CO  0 

»n  0  T  X  Ci 

m  X  CO  ^  1-1 

£-  C:  to  C5  I- 

O  ci  c;  ci 
'3'«nt-C5 

C-.  C-.  X  i-l- 
■r-  d  l-  {~  C5 

CO  .-  CO  OJ  o 

—  CO  •--:  i  -  ci 

0}  7>  7)  1J  T> 

X  L-  CO  0  I- 
0  ":•  -r  -o  i- 

CO  CO  CO  CO  CO 

CO  cc  uo  0  in 
c:  0  oj  —  .n 

CO  -r  T  -^  .* 

o-r  X  5J  0 

£-  X  ~.  —  7? 

".-  -r  T  in  in 

CO  ^  3  t*-  X 

in  i-o  in  i-o  »n 

Ch 

.*oxo 

OJ-VOQC  O 

o>  T><  o  00  o 

C>TJ<OXO 

O»rrcoxo 

(NttOXO 

e?  TT  CO  X  0 

00  if5  I-  c:  ■:> 

-roxoco 

CJ  7J  1J  CO  CO 

CO  r;  COT  -r 

CO  X  0  OJ  m 
■^T  1^  in  in  in 

£-  Ci  —  CO  to 

in  m  to  CO  CO 

Sf^eV-J- 

- 

1-I 

oooo 

§§111 

cj  cj  •?;  cj  CO 

00000 
o>-rto  X  0 
CO  CO  CO  CO  •<»• 

o»  —  5  X  0 

■Tf  ^  ~r  -r  •.'. 

1-0  in  in  0  to 

iiiii 

TABLE    XVI.— TRANSITION    CURVE   TABLE. 


36^ 


oo  ooooo 


;— oci~  ooooo  ooooo  ooooo  ooooooo 

•  C*  CO -T  to     Oi-XSSO     »— T*  00 -T  O      «  I- X  Oi  O      —  1»  PI  —  v-  -^  I- 

.»-iT-i»-iT-.    .-•i-i.-ir-iS'*    ©»o»5*ff*T*    o»2*7*~*w    oocc©?ec«orcco 


Si 


too     I- oo  If  1-1      woo  i- CO     i-O'T'—O      O— -TOO      -r-^OOO      -^OXCii-c^O 

^  w    00  o  to  X  o    ij  -J"  i-  o  ■:?    o  ci  T*  o  o    «  x  •:*  to  i-"    to  •-  to  c*  i-    «  x  t  o  i-  cc  o 
.,_(    ,_,__^2j    5*7t?rjoeo    ■^Tooto    cdt-t^xx    cjcsoi— i->?»;» 


H 

OO 

o  -^  o  ■^  o 

o  o  o  o  it 

o  -r  cj  -T  o 

I-  X  X  C5  o 

-rt-X  t^  X  CO 

O  O  —  ^  T» 

X  r? I-  t>  i- 

C^  COCO-T— • 

o  u-  to  to  r^  i-  X 

fe, 

CO  — 

soo 

• 

Tj-oo-rcc 
t-  o  cc  to  o 

1—1 1—1 1—1  *? 

osocci-to 
T  c;  -r  o  o 

Ci  5J  TO  JO  -T 

X  -r  1"  o  i- 

—  X  O  5J  O 
O  O  to  t-  X 

X  d  d  r-'  ?  j 

13.50 
14.01 
15.09 
10.81 
17.97 

d  o  1-^  2*  -r  O  t- 
1-  I»  ?<  ct  ct  <n  w 

> 

V 

« 

* 

X  — TJ<X  CO 

1-"  1-1  i-c  C> 

OJ  CO  T  O  O 

«C<3-^»0» 

t-xcsow 

«  5»  (N  CO  00 

•^lOt^C5i-i 

1— •  1— ( 1— "  1— <  5? 

CO  ■^'*i9'  O 

00  CO  00  1—  -^  t--  o 
oioicicocrivi  •*' 

oo  tOcDl-XCl 

X 

» 

2.93 
3  99 
5.20 
0  57 

8.11 

9.80 
11.05 
13  00 

15.83 
18.15 

oj  -r  ~>  o  iJ 
CO  -:>  o  o  o 

o  CO  -.o  X  ■:> 

(NIJ-IJ  TJCO 

O  —  5>  i-  to 
0?  to  1-  I'  o 

o  X  •:?  o  d 

CO  COT-* -^ 

C5  O  O  t-  CO 

T  O  I-  O  O 

CO  I-  —  to  d 
oooot- 

„ 
H 

OC5 

o'ci 

CC-V 

X  X  I-  o  ■<»• 
O  O  t-  X  ci 

c»oi--<ro 

to  ^  O  O  M 

CO  I-  X  CS  00 

t-  o  oj  X  CO 

t-  O  5»  CO  0»  —  Ci 

O  ~.  X  X  X 

i-  i-  tr  1.-:  o 

O  O  i-X  o 

ct  cj  s<  oi  55 

ox«-o-r 

T  o  to  t-  X 

cj  ;<  sj  7»  7J 

OJ  -^  O  t--  O  ??  o 
si  CO  «  55  CO  CO  « 

o 

too 

T«  X  CJ  CO  o 

-S"  X  5>  O  O 

I*  00  ©J  o  o 

rrX5*cOO 

's-xoitoo 

■*xc»coo~x 

O  C- 

X  O  —  '7>  •«*• 

o  to  X  o  — . 

•— ii-i^i-i  w 

CJ  CO  o  to  X 
CN  ;j  TJ  0<  5» 

o  o  ■:)  CO  <n 

C<  CO  CO  CO  CO 

CO  t-  O  O  5? 

COCOCO-V-Si 

CO  '^  to  t,  o  o  — 

■W  ^  -T  -^  O  O 

Ss 

r-  0» 

•«*<tOOOTt> 

CO  -r  to  I-  C5 

T  O  X  ij*  1-1 
^  CO  O  X  1-1 

.^"  ^  ^  .^  5» 

oo  wo  — 

•^  i-  O  CO  I- 

c»  ;j  CO  CO  CO 

c:  X  cs  —  o 

o  T  X  CO  I- 

■T  Tf  IS-  o  O 

CO  to  I- 1-  X 

p  k"!  r  2E  £r  *-  3c 

{-  0?X  CO  o  o  — 

a6c^.c^.~i6^  oi 
1— 1 1— '  1— 1 1— 1 

H 

OO 
3*  tJ 

o  oo 

o  o  d  -r  o 
CO  CO  CO  ■^r  "T 

O  X  X  I-  t- 

O  O  O  O  I- 

oooTreo 

d  -c"  d  t'  d 
t-  X  X  o  o 

104.2 

109.1 
113.9 
118.8 
123.7 

128.5 
133.3 
138.1 
142.9 
147.7 

^5*05  1-  — 1-X 

o>  f-^  — '  to'  —  to  d 

o  o  to  to  I-  i-  X 

^ 

o  t- 

X  CO  —  CO  o 
CO  O  CJ  O  X 

1—  1— •  1— • 

X  —  X  OTt< 

?*  I-  —  to  ■?> 

ci  •:»  r:  CO  t 

0»  T  05  X  O 

X'  -?■  o  I-  o 

•V  o  o  to  I- 

8.26 

9.00 

9.90 

10.77 

11.07 

1-  o  o  o  CO 

CO  o  to  O  I- 
C?  CO  -^  o  o* 

17.85 
19.00 
20.18 
21.. 39 
22  01 
23.92 
25,23 

> 

5 

0 

—  OJ  CO  o 
O 

to  O  ?J  o  o 

T-  .^   T-( 

C?  CO  ??  -*  o 

T^  O  to  C^  1-^ 
C*  CO  CO  —  o 

coxes  1—  CO 
1-*  1-H 

CO  i^  X  o  o 
1-1 

Ci  ?»  C»  C»  Oi 

^  CO  O  O  X 

TO  CO  CO  ^  ^ 

C  7>  T- to  O  ^  -f 

Si  5J  si  si  si  00  CO 

O  O  CO  to  t-  t-  I- 

^ 

1.21 
1.89 

2  72 
3.70 
4  83 

—  o 

1-^ 

X 

7> 

?•  X 
i-X 

X 

1— « 

-.  ^.  ~.  ^. 
ij  ;*  5/  5i 

32  81 
35.95 
39.23 
42.01 
40.19 

49.87 
53.07 
57.00 
01.00 
05.85 

CO  X  0»  X  O  —  -T 

T-.  O  1-  i~  O  -f  -T 

d  —  o  CO  X  CO  X 

i-  L-  t-  X  X  O  C-. 

1    « 

oo 

O  d 

O  XI- 

o  o 

X  ci 

CO 

o 

i 

X 

to  CO 
X  X 

o 

1— 

o  o  o  o 

;- 1-  to  i~ 
to<.-x  o 

1— •  —  ^^  »— 1 

CO  to  X  o  ■— 
o  -^  r?  CO  ■:» 

O  —  ''  CO  ■•?• 

»-  O  O  i-  T 

i-'dx'<-d 
o  to  to  i-  X 
St  SI  St  St  SI 

O  O  O  CO  O  t-  X 
O  r^  C»  O  X  to  -rf 
5j  f?  CO  CO  00  CO  CO 

o 

CIO 

o  to' 

X  — — 

t-o 

^-  CO 

CO 

1—* 

t4 

^^ 

TJ  O 
X  O 

X 

55 

—  •Wl-O 

*>  CO  "—  to 

si  St  cj  ci 

CO  o  o  ;>  o 
t-  X  d  —  -> 

Cl  3*  Si  CO  CO 

X  —  -r  I-  o 

00  CO  CO  CO  CO 

CO  to  O  C>  O  X  — 

O  —  7»  -•  1-  t;  X 

ijj-  1J1  -VT  T  -^  -V 

OO    ooooo    ooooo 

TO     to  i-  X  O  C:     T-  o>  ?r  —  1-: 


oo  ooooo  ooeoo  ooooo  ooooo  ooooooo 

c.  ~    T-  o>  ?r  —  o    tr  . -  X  c:  o    —  '^.f  co  -r  o    to  i-  x  o  o    —  ■:»  rr  —  i-  to  jt 

^    1 1-  ~-  •-     ^  —  ^  .-.  c>    c»  S»  O*  01  SJ    St  St  SI  S>  T-:     r?  r?  rr  rr  rr  -:  c-- 


368 


TABLE   XVI.— TRANSITION    (TRVE   TABLE. 


•-« 

ooo 

M  ■^  O 

ooooo 

;^  i-  ao  oi  o 

ooooo 
—  ?>  CO  -fin 

ooooo 

ooooo 
1—  OI  CO  •n-  in 

OI  01  01  01  OI 

ooooo 

CO  i-  X  35  O 
Ct  Ol  7^1  01  CO 

OOOOOO 

—  01  CO  -T  in  CO 

'?0  CO  9Q  00  JO  CO 

> 

o 

o 

"b) 

i^r>u 

I-  T*<  -JO  L-J  O 
•"J-  O  3D  O  03 

t-  O  GO  C»  Oi 

in  00 .—  in  GO 

QCC5OJ00  5O 
0»  O  i-c  O  O 

CO  X  OIX  CO 

in  o  CO  1-1 1-. 

CO  t^  o  m  OI 

CO  35  CO  0IO5 

o  o  o  CI  CO  — 

O  CO  O  I-  1*  01 

1-H  f— 1 

^  i-iO?  Ol  <?J 

CO  CO  "^  ^  in 

in  CO  CO  t-  C'l 

XXOiOO 

1-1 1-1 

1-1  OI  CO  CO  Tt  m 

« 

oocsoo 

CO  t-  o  in  T»< 

00  5»  O  35  I- 

in':i<z>t^in 

^X'j<^t- 

OI  X  CO  X  01  CO 

O-TCi-f  05 
CO  CO  CO  "T  "T 

T  05  ^^  35  •I' 

in  o  CO  <»  I- 

05  T»<  05  00  GO 

{-  CO  00  35  35 

CO  X  CO  t-  OI 

OO' 01 

{-  r-1  to  1-1  O 
01  CO  CO  Tfi  iji 

O  Tt  35  00  X  0? 

in  m  m  CO  CO  t- 

fc. 

"*  00  t-  ^^  -< 
C5  ■?»  O  i-H  to 

in  o  cs  05  -^ 
^  I-  00  o  X) 

eO  GO  GO  o»  •— 1 

eo  T  COCO  CO 

in  ■-<  X  t-  o 
CO  -*"  in  J.-0 

f-  05  CO  t-  OI 

01  in  05  CO  X 

i-iinci-f  oo 
CO  X  ■«•  o  I-  •* 

1-1 '-TJC* 

CO  CO  -^  in  o 

O  £.-  00  35  O 

i-i  01  CO  ■*  CO 
1—1  1— H  i-H  1—1  1—1 

t-  X  05  •rt  01 
1-1 1-1 1-1  01  (M 

■»r  in  t-  05  o  01 

CI  CI  01  CI  CO  CO 

« 
V 

O 

?o*j  GO  inrfi 

^  C«  W  CO  T>< 

msot-oso 

01  Tf  CO  X  o 

C0C0  05  0J10 

OS  CO  t»  1-H  CO  1— • 

i-i^i^oi  CO 
o 

Tfinooco 

T-C 

»-<  1-1  OI  <w  ?» 

1—1 1—1 1— 1 1—1  01 

eo  00  CO -«*•  Tji 

(MOKMOOOO 

in  in  CD  CO  t-- 

eo  •^  T^  o  in  CO 

X  X  05  O  1— 1  01 

1—1  1-H  1-^ 

fe 

•^  t-  -^ 

o -Hi- coo 

O  ?>  GO  •»*<  05 

•n  35  -*  OJ 1-" 

S-Si2g?J 

m  i-  CO  01  01 

t-  35  00  05C0 

00  O  I-  X  CO 

IT  in  coo  •* 

CI  CO  1--  Tt  -r  in 
o  t-  m  m  -o  X 

—IIJ 

CO  irsoaoo 

1^ 

CJ  "*  t-  O  CO 

1-  .-H-H  »?  ?{ 

«  O  CO  oo 
0»  ?i  CO  CO  -^ 

TT  X  CO  i-  OI 

TT  ir  m  in  CO 

t-  It  i~  -it  X 

CO  I- 1-  X  X 

^■oini-f-co 

05  05  O  —  1-1  01 

r^ 

H 

OCJ 

QO  e-  •«  CO  o 

t-  CO  X  CO  i- 

O  OJ  CO  CO  r» 

O  I-  CO  I-  o 

OI  01 1-  X  -T 

05  CI  CO  CI  o  t- 

00  05 

c;  C5  s;  05  C5 

O  O  I-  00  Oi 

GO  30  i-  I-  O 

O  -^OJCO-J* 

to  in  -r  CO  o> 

m  O  {  -  30  35 

1—  35X  CO  in 

OI  OI  01  01  01 

CO  —■  35  CO  T)" 

■T  in  in  CO  I- 

(?!  O)  01  II  OI 

—  05  CO  CO  O  CO 
XX050  —  — 
CI  OI  CI  CO  CO  CO 

a 
•-1 

■^C-JO 

ooOtJi  wo 

GO  so 'T 'MO 

GO  --0  -*  o»  o 

X  CO  -T  01  o 

X  CO  TT  01  o 

xcotjioiox 

irt  £-OS 

O  ?»-><«>  GO 

^^   .-1  7^  I-H   .-1 

05 '-'  CO  in  i- 

— .  OJ  T«  7»  o» 

00  O  01  -^  CO 

oi  CO  00  CO  00 

t-o>  rt  CO  in 

CO  CO  i1»  TT  ^ 

CO  X  o  OI  T 
•^  ■T  lO  in  in 

•n  i-  35  —  CO  IT 

•n  in  in  CO  o  CO 

> 

as 

a 
O 

o 

C3 

?» 

ooocs 

.-1  1-c  0* 

OJ  e-  rp  Tf  «o 
r("  O  <-  05  i-H 

05:0Tf  irtGO 
00  O  05  1J  o 

C0005000 

35  CO  ;o  ^  in 

X  in  -rti  in  i~ 

05  1*  05  -^05 

CI  X  CO  in  CO 
in  o  CO  01 X 

X  01 1-  t  01  01 

1*  —  I-  •«•  1-  X 

»-i 

■•-1  T^  y-l  7t  Qi 

0?  CO  CO  T}<  -"l" 

mnmcoco 

t-  X  X  O  35 

O  1-1  -1  01  CO  CO 

H 

lO  o  »o 

05  Ot  05  Oi 

00  00 1~- to  m 

r*<  CO  0?  1-  OS 

X  CO  -cf  01  o 

t-  in  01 35  in 

CIXr)"0  in  1-1 

O  -fOl  -^05 
CO  coco  rr  Tf 

•rC  35  •*  35  'S" 

in  m  o  o  I- 

35 -f  35  1*00 

t-  GO  GO  35  C5 

ooxcoxco 

O  O  rH  —  01 

{,  '>j  J,  ,-,  o 

OI  CO  CO  IT  -rr 

—  in  o  in  35  -»i 
in  in  CO  CO  CO  t- 

fe. 

c»  «>  o 

rf -^  Oi  <x>  irt 

ao—cTfooco 

00  coo  in  ?j 

—  t-O-HO 
C^  O  -T  CO  01 

CO  O  01  3S  35 
—  --  ^  .-■  01 

CO  OI  -t  1-1  OI 

•»■  CO  X  i-c  -f- 

CO  m  t-  rr  -^  00 

t-  1-1  in  o  ino 

•-I  rl  T-l  W 

o»  CO  CO  -'t  in 

in  ;o  t-  00  OS 

O  1-1  01  CO  Tt< 

m  CO  t^  35  o 

1—   1—   1-1  ,-c  01 

—  CO  Tf  CO  I-  35 

CI  0?  01  CI  CI  01 

"b 
« 

o 

:^<  00  »n  I-  o 

00  1-  -H  lO^H 
— 1  —  0»  OI  CO 

TflOCDt-CO 

OS^OITTCO 

X  o  CO  m  X 

— <  Tji  I- 1-1  in  35 

y->  T-i  T^ /a -it 

o 

00  ■*  in  t-  GO 

»-i  1-1 1-1  OJ  01 

1-^  1—1  t— 1 1— 1 
0?  CO  CO  CO  ^ 

1-1  01  OI  01  OI 

CO  CO  CO  t  T  IT 

CO  r-  X  X  35  o 

~ 

t-i 

o  lO  CO  o  in 

CO  in  35  o  1?/ 

GO  35QO  -KGO 
^  C?  in  O  50 

05  1-  0?CO  ^ 

in  m  1—1 01  o 

O  X  X  35  1-1 

o  m  m  35 1- 

CD  —X  CO  CO 

X  —  x«-x  o 
t-  o  CO  X  t  0) 

•^^» 

cT*  in  I-  C5 

—  CO  in  GO  o 

Tl  — i  I-H  T-H  01 

CO  CO  35  ?>  O 

CI  OI  01 00  00 

O  CO  {-  —  CO 

Tji  1*  iji  in  in 

O  lO  35  ^r  35 
CO  CO  CO  I-  l"- 

f-1 1^  1— ' 

5? 

05SS 

GOt-tO-T'?? 

(35  35  O:  35  C5 
»n  O  I-  33  35 

O  J-  CO  05  Tj< 

35  :o  30  I-  I- 

O  —  1J  CO  -T 

00  OI  in  t-  30 

Co'  --o'  lO  -*•  CO 
•n  CO  t-  30  35 

35  X  I-  -T  —1 
OI  1-1  O  35  30 

0  --  OI  01  CO 

01  01  01  01  01 

COO  CO  in  CO 

CO  in  CO  —  35 

't  m  CO  J-  t- 
o»  01 01  OI  c^t 

in  ■»#  1-1  CO  o  CO 

t;J  in  00  o  x  in 

X  o5  o  ^  —  01 

CI  01  CO  CO  CO  CO 

0 

GCrt-O 

O  -O  CO  rr  o 

O  0»  30  tT  o 

CO  01  X  ■*  o 

--0  01  X  -r  o 

CO  0>XTf  o 

CO  01  X  Tf  O  CO 

Tf-OOO 

05  T-i  1?  Tt"  :0 

C>»  35  O  ■?»  T»< 

.-Hl-H  0»  OI  <?* 

in  {-  X  o  o» 
OI  OI  OI  00  ?3 

coincoxo 

CO  CO  CO  CO  ^ 

—  00  It  cox 

TJl  -IJ"  Tf  TT  t 

05  1-hoi  t  CO  ^- 
1r  in  in  in  m  «n 

<*» 

OOO 
r?  TO 

ooooo 

40  r.-.  GO  c:  O 

OOOOO 

—'  ■?»  CO  -T  m 

ooooo 

ooooo 
—  OI  CO  -^  in 
oj  01  01  r:i  01 

OOOOO 
CI  01  01  CI  00 

OOOOOO 

»-  01  CO  TTlft   CO 

CO  CO  CO  00  CO  CO 

TABLE   XVI.— TRANSITION    (TRVE   TABLE. 


3G9 


f-w 

o  oo 
CO  ^  «n 

c:  o oo o 

«3  I-  3C  Oi  O 

ooooo 
•^  1*  TO  -T  m 

ooooo 

50  i  -  X  OS  o 

1— 1  1— t  -r^  1— •  C*i 

ooooo 

— .  fW  CO  -r  lO 
<MO»OJO«Ct 

W  O*  «  SJ  CO 

OOOOOO 
—  O*  CO  T^  lT  O 
CO  CO  CO  CO  CO  CO 

> 

a 

o 

o 

5 

"Si 

-too 

t-  00  <?»  00  GO 

»n  i"  o  sj  if3 

ooooo 

OS  C-J  o  o  o 

I-  CO  <^  «  CO 

OS  -a*  OS  o  o 

J-Xi—  oco 
O  CJ  OS  o  c^ 

CJ  <?>  O  GO  CO 
OSOCOOX 

OS  {-  o  -t  -t  in 
in  CO  1^  OS  I-  o 

»-»  T-^  T— « 

^oj  wcoco 

COiJi-^OO 

O  I-  t~  00  OS 

OS  O  --H  'TJ  ■?» 

CO  Tt  o  o  o  J- 

H 

C5  0>  C5  GfJ  l- 

I-  o  o  CO  <r» 

oooo  coo 

t-  Tt  o  o  c< 

oo  CO  00  CM  CD 

OS  oj  o  X  o  1-1 

05  -t  05  ^  Oi 
CJ  CO  CO  ■<*>  T 

-^  OS  -r  OS  th 
in  in  o  o  I- 

OS  CO  00  CO  00 

1^  CO  00  OS  C3S 

O  t-  C*  O  ^ 

oo  i-H  rH  0< 

O  O  -t  OS  CO 
0>  coco  CO  ■» 

J-  C*  O  O  ir  OS 

■t  o  o  o  tc  o 

&H 

(MOOD 

1-1  »n  o  lO  '-I 

■^  ?o  in  --  i-H 

oo  O  CO  (W  i-H 

oooc!seooj 

O  1-1 1-1  CO  o 

00  OS  O  I-  -*< 
i-  O  -TOOCO 

t-  O  i-  Tf  CD 

oo  -a-  O  I-  -T 

00  o  o  o  -t>  -r 

OJOCJSX  i-i- 

tH  rt  0»  TJ  CO 

eo  -^  o  o  t- 

00  OS  o  —  o? 

1— » 1— « 1— « 

CO  O  CO  l-  OS 
1—1 1—1 1—1  1—1 1—1 

o  cj  -t  o  I- 

OiOiOtOi  ot 

OS  ^  o>  -rt  O  X 
0»  CO  CO  CO  CO  CO 

"^ 

V 

T— 1 

o 

COTOOCOOO 

O  W  1-1  (?J  o 

o»eo-*oo 

Qoosi-ceoo 

QOi-1  TjlX  1-1 

lO  OS  CO  00  CO 

OS  o  1-1 00  in  CO 

•^  .-1  CJ  CO  T 

o 

o  00  o  CJ  Tt< 

1-1 1-1 1-1 

0?  (?J  (7J  0?  CO 

1-1  OJ  CJ  C4  CO 
Tj<  Tji  ooo 

CO  CO  1^  i^*  in 
I- 1-  00  OS  o 

O  O  I-  I-  X  OS 
.-1  C4  CO  Tt  o  o 

T-l 

1-1 1-1 1-^  1-^  1-1  ^ 

;a 

lO  -*  05 
T-H  O  T-H 

05  -^  -rooto 
in  01  .-•  c*  o 

«»  o  o  t^W 

(M  r-  C^  O  1-1 

OS  X  I-  t-  I- 

ODXOrrO 

l-O-IJOi- 

oocoocooo 

o  X  o  I-  e* 

O  CO  CO  -f  I- 

OSGOOOXXl- 

o  in  rix  o  o 

l-H(JiCO 

rfOOOOW 

in  CO -^  -TOO 

1-1 1-1  0*  i?J  c» 

Ti  O  O  "^  OS 
CO  CO  "^  *^  T}< 

eox-i>os-t< 
in  o  o  o  L- 

ooo*  GO -r 

cocoes  OS  o 
1— < 

1-  I-  Tf"  O  I-  -t 
1—1  T-l  OJ  CO  CO  ^ 

H 

05  Ci  00 

t-  io  c»  05  in 

OOOOOi-H 

1-1  OS  o  i-H  in 

1>X  ?>-^os 

w  -t  cooo 

00  OS  00  Tf  ooo 

O  05  05  00  GO 
in  CD  L-  00  05 

00  4-  O  Ol  O 
O  -^  CJro  TT 

-r  ?f  —  o  00 

O  O  i-QOCO 

CD  -f  <N  O  i- 
OS  O  —  '?>  'T' 
1-1  OJ  O*  CJ  Ci 

O  C8  OS  O  01 

CO  -r  -f  o  o 

oo -rt"  O  CD  ^  t- 
O  I-  X  X  cs  OS 
OiO*0tOt  01  ot 

^H 

<£coo 

ojTfoooo 

ej  Tj.  o  00  o 

OJ-tOXO 

oj  -to  ooo 

C?  Tt  O  00  o 

0*  It  o  X  o  o? 

OQO'- 

M  in  t-  05  c* 

T-i  1— 1  T-t  f— t  Oi 

c»Oic<eoeo 

O  I-  C3S  ^  -+■ 
CO  CO  CO  Tf  ■'3' 

o  00  o  w  in 

TT  "^  O  O  O 

1-  C5S  1-1  CO  O 

inocooo 

00  o  c>  1 1-  fife 

O  t-  J>  I-  i-  i- 

o 

>S 

2§5^ 

o?  --  CO  t--  ■* 
in  i-  05 1—  ** 

Tl<Oi-iOS  OS 

I-  o  m^  1-1 

CJ  GO  O  O  00 

CO  o  o  o  o 

OJ  OS  oooo  T-l 

1-i  o  (?*  00  o 

in  — <os  oso 

1-1  X  IT  1-1  OS 

CO  {-  .rj  X  o  o 

OCO  ^XO  -T 

1— t  T-« 

Tictof  Qtn 

CO  •<*■■*  o  o 

O  O  l^  I-  30 

OSOsO-'-H 

oj  00  -*  -r  o  CD 

H 

looin 

Ci  Oi  OS  00  QD 

i>  i-  in  ■<*  CO 

0*  o  00  o  Tt« 

i-iOOO  N05 

O  1—  O  C-f  t^ 

i-iooeoo  OS 

C:i  f  OS  Tf  05 
C^  CO  CO  tT«  "^ 

■^  o>  -ros  -r 
in  »n  o  o  i>- 

cjj  Tt  xeooo 
I-  00  00  OS  OS 

CO  t-  01  t-  — 

O  O  —  ^  Of 

(N  coco  rr  "!r 

CSCO  00  0>O  o 

-r  o  o  o  CD  t- 

fc. 

?D«oeo 

rfMtomos 

O  ■^00  CO  00 

o  o  I-  in  GO 
o  ^ooo-^ 

oooscoco 

CO  CO  C*  CO  -3< 

00  X  CO  CO  00 
o  t-  o  CO  o 

00  00  CO  00  t- 

o  o  o  o  1-1 

oooo«>aoco 

oo  Tf  Of  C3S  i-  o 

1-1 1-1  t-hW  C* 

CO  ■*  rj"  oo 

t-  X  OS  O  -H 

1—1    T~i 

CJ  CO  O  O  I- 

OS  O  CJ  CO  lO 
T-l  CJ  Oi  o*  o* 

O  X  O  —  CO  o 
Ci  T>  00  CO  CO  CO 

o 

CO  in  L-  ^  in 

1-<  T-l 

0{-  o  Tfco 
Ti  C»  CO  -rr  o 

eooDO^co 

in  i-  o  oi  o 

00  oj  o  o  Tt 

oscox  -t  o  o 

•.-n-cjeo-* 
o 

inooooeo 
1-1 1-1 

T^  T-»  1^ 

©»©JO»©»eo 

TH  i—  5<  •?{  Q} 

e0rt<-<*'0  o 

04  CO  CO  -S"  ■<*• 
o  o  i^  00  OS 

Tf  O  O  O  t-  i- 
OO  —  OJCO  -r 

;a 

O  00  o> 

i-  tC  O  «0  CO 

1-1 53  ■*  com 

CO  -*  i~  <?>  I- 

OS  O  CO  -^  o 

CO  OS  OO  o 
1-  l-O  i-  OS 

CO  OS  J  -  CO  o 
CO  OS  i- I- 00 

Tj<  00  00  ■?>  OS 
1-1  O  ^  OS  I- 

o  Tfo;  o  CO  —1 
X  OS  ^  o  o  o 

«»-l^> 

tP  in  L-  05 '-' 

CO  O  OS  T»  O 

1-1  1-1  r-r  (?J  W 

OS  o>  o  o  -# 
04  CO  CO  ^  ^T 

OS  CO  00  CO  oo 
i^i  o  ooo 

-?■  OS  in  o  CD 

V-  ..-  00  OS  OS 

OJX  Oi-lX  -* 

o  o  1-1  oj  ■^J  CO 

1—1 1—1 1—1 1— 1 1-1 1— 1 

05CO 

I-  O  T)<  l-H  00 

^T  OS  CO  l>  OS 

i-ii-iox  -rr 

O  -*  O  I-  o 

itO'^Ot- 

O  CO  00  1-1  o»  OJ 

oojoj 

CO  CO  "^ 

05  OCJ5  ooo 

in  o  i-  CO  OS 

00 1^  I-  o  o 

O  1-  0>  CO  -# 

1—1   1— 1   1— »    T— 1   1— « 

O  "*  CO  1-1  o 
O  Ot-X  CTS 

T— »   1-1   ^-*   1— I   1— • 

OS  i  -  O  CO  ■— 1 
OSO  — CJ  CO 

osi-  -r  «oo 

CO  -T  O  O  CD 

01  ot  o»  rj  Qt 

O  OfOO  O  1-1  {- 

l-OOOOOS  o  o 
OJ  OJ  OJ  OJ  CO  CO 

e- 

ooo 

ooooo 

COOOO 

ooooo 

OOCOO 

ooooo 

oooooo 

i:^  00  o 

©»  -»<  «0  QC  o 

*f  -:»<  O  X  O 
CI  Ct  C<  Ol  CO 

CJ  TT  O  GO  O 
CO  CO  CO  CO  ■^ 

CJ  rji  O  X  O 

■<!(•  IT  13'  Tt  O 

0)  I*  o  X  o 
ooooo 

0»  -t  O  CO  O  OJ 
CO  O  CO  O  I-  c- 

•^^ 

ooo 

CO  -T  o 

ooooo 

«0  I-  X'  OS  o 

OOOOO 
1-i'N  CO-t  O 

O  l-XOs  o 
1— t  1— 1  rl  1— 1  C4 

OOOOO 
1-1  <?J  CO  t  in 
a  cj  (T*  c*  c* 

ooooo 

O  i-  X  C3S  O 
0»  W  OJ  0*  CO 

ooooo  o 

—  0>  CO  -»  o  o 

CO  CO  CO  CO  00  00 

370 


TABLE   XVI.— TRANSITION    CURVE   TABLE. 


f^ 

c^  rr  in 

OOOOO 

CO  I-  00  C5  O 

OOOOO 
!-•  1*  CO  -r  in 

8R? 

> 

o 

0 

§ 

o 

o 

o 

in 

ooo 
CO  T  in 

ooooo 

OI-00C5O 

1-H 

ooooo 

—  O^CO-^iO 
*-H  1— •  »-^  *-^  ^^ 

ooo 
~  l~  3J 

> 

o 

o 

in 

V. 

O)  'T?  O  ■•-"  CO 
CO  C5  C-J  O  CO 

-^  in  o  00  C5 
c?  CO  ^  in  o 

CD  "I- 

^«.^ 

00  O  I-  CO  CO 
I-  O  CO  I-  ^ 

■  "■  -^  «  c> 

t--i<  -^x  in 
in  o  in  o  o 

0»  CO  00  TJ>  -^i 

ociin 

sjx  in 

in  in  CO 

o 

^-  ^H  »— 1 

c»  c*  eo  eo  Tp 

•^tDkO 

H 

ino  in 

O5Cft00i>-CO 

in-^c^^oo 

coeoo 

C5 

•no  ■* 

05  X  X  I-  in 

Tj<o»ox>n 

0»XTf 

o;  -f  Oi  -r  ci 

C»  CO  CO  Tj<  T 

Tf  C5  TJ-Oi  CO 

in  in  CD  CO  I- 

00  "0  00 

i-QOOO 

05  rr  05  -r  05 

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TABLE  XVII,— AREAS  OF  LEVEL  SECTIONS. 
Base,  26  =  14  feet.    Side  slopes  1}^  to  1. 


371 


C.  H. 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.7 

.8 

.9 

0 

0.0 

1.4 

2.9 

4.3 

5.8 

7.4 

8.9 

10.5 

12.2 

13.8 

1 

15.5 

17  2 

19.0 

20.7 

22.5 

24.4 

26.2 

28.1 

30.1 

32  0 

2 

34.0 

36.0 

38.1 

40.1 

42.2 

44.4 

46.5 

48.7 

51.0 

53.2 

3 

55.5 

57.8 

60.2 

62.5 

64.9 

67.4 

69.8 

72.3 

74.9 

77.4 

4 

80.0 

82.6 

85.3 

87.9 

90.6 

93.4 

96.1 

98.9 

101.8 

104.6 

5 

107.5 

110.4 

113.4 

116.3 

119.3 

122  4 

125.4 

128.5 

131.7 

134.8 

6 

138.0 

141.2 

144.5 

147.7 

151.0 

154.4 

157.7 

161.1 

164.6 

108.0 

7 

171.5 

175.0 

178.6 

182.1 

185.7 

189.4 

193.0 

196.7 

200.5 

204.2 

8 

'J08.0 

211.8 

215.7 

219.5 

223.4 

227.4 

231.3 

235.3 

239.4 

243.4 

9 

'.'47.5 

251.6 

255.8 

259.9 

264.1 

268.4 

272.6 

276.9 

281.3 

285.6 

10 

290.0 

294.4 

298.9 

303.3 

307.8 

312.4 

316.9 

321.7 

326.2 

330.8 

11 

335.5 

340.2 

345.0 

349.7 

354.5 

339.4 

364.2 

369.1 

374.1 

379.0 

12 

384.0 

389.0 

394.1 

399.1 

404.2 

409.4 

414.5 

419.7 

425.0 

430.2 

13 

435.5 

440.8 

446.2 

451.5 

4,56.9 

462.4 

467.8 

473.3 

478.9 

484.4 

14 

490.0 

495.6 

501.3 

506.9 

512.6 

518.4 

524.1 

529.9 

535.8 

541.6 

15 

547.!; 

553.4 

559.4 

565.3 

571.3 

577.4 

583.4 

589.5 

595.7 

601.8 

16 

608.0 

614.2 

620.5 

626.7 

633.0 

639.4 

645.7 

652.1 

658.6 

665.0 

17 

671.5 

678.0 

684.6 

691.1 

697.7 

704.4 

711.0 

717.7 

724.5 

731.2 

•18 

738.0 

744.8 

751.7 

758.5 

765.4 

772.4 

779.3 

786.3 

793.4 

800.4 

19 

807.5 

814.6 

821.8 

828.9 

836.1 

843.4 

850.6 

857.9 

865.3 

872.6 

20 

880.0 

887.4 

894.9 

902.3 

909.8 

917.4 

924.9 

932  5 

940.2 

947.8 

21 

955.5 

963.2 

971.0 

978  7 

986.5 

994.4 

1002.2 

1010.1 

1018.1 

1026.0 

22 

1034.0 

1042.0 

1050.1 

1058.1 

1006.2 

1074.4 

1082.5 

1090.7 

1099.0 

1107.2 

23 

1115  5 

1123.8 

1132.2 

1140.5 

1148.9 

1157.4 

1165.8 

1174.3 

1182.9 

1191.4 

24 

1200.0 

1208  6 

1217.3 

1225.9 

1234.6 

1243.4 

1252.1 

1260.9 

1269.8 

1278.6 

25 

1287.5 

1296.4 

1305.4 

1314.3 

1323.3 

1332.4 

1341.4 

1350.5 

1359.7 

1368.8 

26 

1378  0 

1387.2 

1396.5 

1405.7 

1415.0 

1424.4 

1433.7 

1443.1 

1452.6 

1462.0 

27 

1471.5 

1481.0 

1490  6 

1500  1 

1509.7 

1519.4 

1529.0 

1538.7 

1548.5 

1558.2 

28 

1568.0 

1577.8 

1587.7 

1597.5 

1607.4 

1617  4 

1627.3 

1637.3 

1647.4 

1657.4 

29 

1667.5 

1677.6 

1687.8 

1697.9 

1708.1 

1718.4 

1'728.6 

1738.9 

1749.3 

1759.6 

Base,  26  = 

=  15  feet.     Side  slopes  1}4  to  I. 

C.  H. 

.0 

1 

o 

3 

4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

1.5 

3.1 

4.6 

6.2 

7.9 

9.5 

11.2 

13.0 

14.7 

1 

16.5 

18.3 

20.2 

22.0 

23.9 

25.9 

27.8 

29.8 

31.9 

33.9 

2 

36.0 

38.1 

40.3 

42.4 

44.6 

46.9 

49.1 

51.4 

53.8 

56.1 

3 

58.5 

60.9 

63.4 

65.8 

68.3 

70.9 

73.4 

76.0 

78.7 

81.3 

4 

84.0 

86.7 

89.5 

92.2 

95.0 

97.9 

100.7 

103.6 

106.6 

109.5 

5 

112.5 

115.5 

118.6 

121.6 

124.7 

127.9 

131.0 

134.2 

137.5 

140.7 

6 

144.0 

147.3 

150.7 

154.0 

157.4 

160.9 

164.3 

167.8 

171.4 

174.9 

7 

178  5 

182.1 

185.8 

189.4 

193.1 

196.9 

200.6 

204.4 

208.3 

212.1 

8 

216.0 

219.9 

223.9 

227.8 

231.8 

235.9 

239.9 

244.0 

248.2 

252.3 

9 

256.5 

260.7 

265.0 

269.2 

273.5 

277.9 

282.2 

286.6 

291.1 

295.5 

10 

300.0 

304.5 

309.1 

313.6 

318.2 

322.9 

327.5 

332.2 

337.0 

341.7 

11 

346  5 

351.3 

350.2 

361.0 

365.9 

370.9 

375.8 

380.8 

385.9 

390.9 

12 

396.0 

401.1 

400.3 

411.4 

416.6 

421.9 

427.1 

432.4 

437.8 

443.1 

13 

448.5 

453.9 

459.4 

464.8 

470.3 

475.9 

481.4 

487.0 

492.7 

498.3 

14 

504.0 

509.7 

515.5 

521.2 

527.0 

532.9 

538.7 

544.6 

550.6 

556.5 

15 

502.5 

568.5 

.574.6 

580.6 

586.7 

592.9 

599.0 

605.2 

611.5 

617.7 

16 

624.0 

630.3 

6.36.7 

643.0 

649.4 

655.9 

662.3 

668.8 

675.4 

681.9 

17 

688.5 

695.1 

701.8 

708.4 

715.1 

721.9 

728.6 

735.4 

742.3 

749.1 

18 

756.0 

762.9 

769.9 

776  8 

783.8 

790.9 

797.9 

805.0 

812.2 

819.3 

19 

826.5 

833.7 

841.0 

848.2 

855.5 

862  9 

870.2 

877.6 

885.1 

892.5 

20 

900.0 

907.5 

915.1 

922.6 

'930.2 

9^7.9 

945.5 

953.2 

961.0 

968.7 

21 

976.5 

984.3 

992.2 

1000.0 

1007.9 

1015.9 

1023.8 

1031.8 

1039.9 

1047.9 

22 

1056.0 

1064.1 

1072.3 

1080.4 

1088.6 

1096.9 

1105.1 

1113.4 

1121.8 

1130  1 

23 

1138.5 

1146.9 

11.55.4 

1163.8 

1172.3 

1180.9 

1189.4 

1178.0 

1206.7 

1215.3 

24 

1224.0 

1232.7 

1241.5 

1250.2 

1259.0 

1267.9 

1276.7 

1285.6 

1294.6 

1303.5 

26 

1312.5 

1321.5 

1330.6 

1339.6 

134S.7 

1357.9 

1367.0 

1376.2 

1385  5 

1394.7 

26 

1404.0 

1413.3 

1422.7 

1432.0 

1441.4 

1450.9 

1460.3 

1469.8 

1479.4 

1488.9 

27 

1498.5 

1.508.1 

1517.8 

1527.4 

1.537.1 

1546.9 

1.556.6 

1566.4 

1576.3 

1.586.1 

28 

1590.0 

1605.9 

1615.9 

1625. 8 

1635.8 

1645.9 

16.55.9 

1606.0 

1076.2 

1686.3 

29 

1696.5 

1706.7 

1717.0 

1727.2 

1737.5 

1747.9 

1758.2 

1768.6 

1779.1 

1789.5 

372       TABLE  XVII.— AREAS  OF  LEVEL  SECTIONS 

Base,  26  =  28  feet.     Side  slopes  U  to  1. 


C.  H. 

0 
1 
2 
3 

4 

5 
6 
7 

8 
9 

10 
11 
12 
13 
14 

15 

16 
17 
18 
19 

20 
21 
22 
23 
24 

25 
26 
27 

28 
29 


.0 


.2 


.7 


.8 


C.  H. 

0 
1 
2 
3 
4 

5 
6 

4 

8 
9 

10 
11 
12 
13 
14 

15 
16 
17 
18 
19 

20 
21 
22 
23 
24 

25 
26 
27 

28 
29 


0.0 

29.5 

62.0 

97.5 

136.0 

177.5 
222.0 
269.5 
320.0 
373.5 

430.0 
489.5 
552.0 
617.5 
686.0 

757.5 
832.0 
909.5 
990.0 
1073.5 

1160. 

1249, 

1342.0 

1437.5 

1536.0 

16.37.5 
1742.0 
1849.5 
1960.0 
2073.5 


.0 
,5 


.0 


2.8 

32.6 

65.4 

101.2 

140.0 

181.8 
226.6 
274.4 
325.2 
379.0 

4.35.8 
495.6 
558.4 
624.2 
693.0 

764.8 
839.6 
917.4 
998.2 

10S2.0 

1168.8 
1258.6 
1351.4 
1447.2 
1546.0 

1647.8 
1752.6 
1860.4 
1971.2 
20S5  0 


5.7 

35.8 

68. 9 

105.0 

144.1 

186.2 
231 .3 
279.4 
330.5 
384.6 

441.7 
501.8 
564.9 
631.0 
700.1 

772.2 

847.3 

925.4 

1006.5 

1090.6 

1177.7 
1267.8 
1360.9 
1457.0 
1556.1 

16.58  2 
1763.3 
1871.4 
1982.5 
2096.6 


8.5 
38.9 
72.3 
108.7 
148.1 
190.5 
235.9 
284.3 
3:3,5.7 
390.1 

447.5 
507.9 
571.3 
637.7 
707.1 

779.5 

854.9 

933.3 

1014.7 

1099.1 


11.4 

42.1 

75.8 

112.5 

152.2 

194.9 
240.6 
289.3 
341.0 
395.7 

453.4 
514.1 
577.8 
644.5 
714.2 

786.9 

862  6 

941.3 

1023.0 

1107.7 


14.4 

45.4 

79.4 

116.4 

156.4 

199.4 

245.4 

294 

:i46 

401 

459 
520 
584.4 
651.4 
721  4 

794.4 

870.4 

949.4 

1031.4 

1116.4 


17.3 

48.6 

82.9 

120.2 

160.5 

203.8 
2.50.1 
299.4 
351.7 
407.0 

465.3 
526.6 
590.9 
658.2 
728.5 

801.8 
878.1 
957.4 


20.3 

51.9 

86.5 

124.1 

164.7 

208.3 
254.9 
304.5 
;357.1 
412.7 

471.3 
5.32.9 
597.5 
665.1 
735.7 

809.3 
885.9 
965.5 


1039.7  1048.1 
1125.0  1133.7 


26.4 

58.6 

93.8 

132.0 J 

173.2 

217.4 
264.6 
314.8 
368.0 
424.  ii 

483. -f^ 

545.6 

610.8 

679.0 

750.2 

824.4 
901.6 
981.8 
10.56.6  1065.0 
1142.5  1151.2 


23.4 
55.3 
90.2 

128.1 
169.0 

212.9 
259.8 
.309.7 
362.6 
418.5 

477.4 
5.39.3 
604.2 
672.1 
743.0 

816.9 
893.8 
973.7 


1186.5 

1195.4 

1204.4 

1276.9 

1286.1 

1295.4 

1370.3 

1379.8 

1389.4 

1466.7 

1476.5 

1486.4 

1566.1 

1.576.2 

1586.4 

1668.5 

1678.9 

1689.4 

1773.9 

1784.6 

1795.4 

1882  3 

1893.3 

1904.4 

1993  7 

2005.0 

2016.4 

2108.1 

2119.7 

2131.4 

1213.3  1222.3  12.31.4  1240.4 
1304.6  1313  9  1323.3  13.32.6 
1398.9  1408.5  1418.2  1427.8 
1496.2  1506.1  1516.1  1526.0 
1596.5  1606.7  1617.0  1627.2 


1699.8  1710.3 
1806.1  1816.9 
1915.4  1926.5 
2027.7  2039.1 
2143.0  21.54.7 


1720.9  1731.4 
1827.8  18.38.6 
1937.7  1948.6 
2050.6  2062.0 
2166.5  2178.2 


Base,  26  =  18  feet.     Side  slopes  1  to  1. 


.7 


.8 


.9 


0.0 
19.0 

40.0 
63.0 
88.0 

115.0 
144.0 
175.0 
208.0 
243.0 

280.0 
319.0 
360.0 
403.0 
448.0 

495.0 
544.0 
595.0 
648.0 
703.0 

760.0 
819.0 
880.0 
943.0 
1008.0 

1075  0 
1144  0 
1215.0 
1288.0 
1.363.0 


1.8 
21.0 
42.2 
65.4 
90.6 

117.8 
147.0 
178.2 
211.4 
246.6 

283.8 
323.0 
364.2 
407.4 
452.6 

499.8 
549.0 
600.2 
6.53  4 
708.6 

765.8 
825.0 
886.2 
949.4 
1014.6 

1081.8 
1151.0 
1 222  2 
1295  4 
1370.6 


3.6 
23.0 
44.4 
67.8 
93.2 

120.6 
150.0 
181.4 
214.8 
250.2 

287.6 
327.0 
368.4 
411.8 
457.2 

504.6 
554.0 
605.4 
658.8 
714.2 

771.6 
831.0 
892.4 
9.55.8 
1021.2 

1088.6 
11.58.0 
1229  4 
1302.8 
l'-78.2 


5.5 
25.1 
46.7 
70.3 
95.9 

123.5 
1.53.1 
184.7 

218.3 
253.9 

291.5 
331.1 
372.7 
416.3 
461.9 

509.5 
559.1 
610.7 
664.3 
719.9 

777.5 
837.1 
898.7 
962.3 
1027.9 

1095.5 
1165  1 
1236.7 
13)0.3 
1.385.9 


7.4 
27.2 
49.0 
72.8 
98.6 

126.4 
1.56.2 
188.0 
221  8 
257.6 

295.4 
.335.2 
377.0 

420.8 
466.6 

514.4 
564.2 
616.0 
669.8 
725.6 

783.4 
843.2 
905.0 
968.8 
1034.6 

1102.4 
1172.2 
1244.0 
1317.8 
1393.6 


9.3 

29.3 

51.3 

75.3 

101.3 

129.3 
159.3 
191.3 
225.3 
261.3 

299.3 
339.3 
381.3 
425.3 
471.3 

519.3 
569.3 
621.3 
675.3 
731  3 

789.3 
849.3 
911.3 
975.3 
1041.3 

1109.3 
1179  3 
1251.3 
1325.3 
1401.3 


11.2 

31.4 

53.6 

77.8 

104.0 

132.2 
162.4 
194.6 

228. 8 
265.0 

303.2 
343.4 
385.6 
429.8 
476.0 

524.2 
574.4 
626.6 
680.8 
737.0 


13.1 
33.5 
55.9 
80.3 
106.7 

135.1 
165.5 
197.9 
232.3 

268.7 

307.1 
347,5 
389.9 
434.3 
480.7 

529.1 
579.5 
631.9 
686.3 
742.7 


15.0 
35.6 

58.2 

82.8 

109.4 

138.0 
168.6 
201.2 
235.8 
272.4 

311.0 
351.6 
394.2 

438.8 
485.4 

5.34.0 
584.6 
6.37.2 
691.8 
748.4 


17.0 
37.8 
60.6 
85.4 
112.2 

141.0 
171.8 
204.6 
239.4 
276.2 

315.0 
355.8 
398.6 
443.4 
490.2 

5.39.0 
589.8 
642.6 
697.4 
754.2 


795.2  801.1 

807.0  813.0 

855.4  861.5 

867.6  873.8 

917.6  923.9 

9.30.2  936.6 

981.8  988.3 

994.8  1001.4 

1048.0  1054.7 

1061.4  1068.2 

1116.2  1123.1 

11.30  0  1137.0 

1186.4  1193.5 

1200.6  1207.8 

1256.6  1265.9 

1273.2  1280.6 

1332  8  1340.3 

1347  8  1.355.4 

1409  0  1416.7 

1424  4  1432.2 

TABLE  XVII. -AREAS  OF  LEVEL  SECTIONS.       373 
Base,  26  =  20  feet.    Side  slopes  1  to  1. 


C.  H. 

.0 

.1 

o 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

2.0 

4.0 

6.1 

8.2 

10.3 

12.4 

14.5 

16.6 

18.8 

1 

21.0 

23.2 

25.4 

27.7 

30.0 

32.3 

34.6 

36.9 

39.2 

41.6 

2 

44.0 

46.4 

48.8 

51.3 

53.8 

56.3 

58.8 

61.3 

63.8 

66.4 

S 

69.0 

71.6 

74.2 

76.9 

79.6 

82.3 

85.0 

87.7 

90.4 

93.2 

4 

96.0 

98.8 

101.6 

104.5 

107.4 

110.3 

113.2 

116.1 

119.0 

122.0 

5 

125.0 

128.0 

131.0 

134.1 

137.2 

140.3 

143.4 

146.5 

149.6 

152.8 

6 

156.0 

159.2 

162.4 

165.7 

169,0 

172.3 

175.6 

178.9 

182.2 

185.6 

7 

189.0 

192.4 

195.8 

199.3 

202.8 

206.3 

209.8 

213.3 

216.8 

220.4 

8 

224.0 

227.6 

231.2 

234.9 

238.6 

242.3 

246.0 

249.7 

253.4 

257.2 

9 

261.0 

264.8 

268.6 

272.5 

276.4 

280.3 

284.2 

288.1 

292.0 

296.0 

10 

300.0 

304.  C 

308.0 

312.1 

316.2 

320.3 

324.4 

328.5 

.332.6 

336.8 

11 

341.0 

345.2 

349.4 

353.7 

358.0 

362.3 

366.6 

370.9 

375.2 

379.6 

12 

384.0 

388.4 

392.8 

397.3 

401.8 

406.3 

410.8 

415.3 

419.8 

424.4 

13 

429.0 

433.6 

438.2 

442.9 

447.6 

452.3 

457.0 

461.7 

466.4 

471.2 

14 

476.0 

480.8 

485.6 

490.5 

495.4 

500.3 

505.2 

510.1 

515.0 

520.0 

15 

525.0 

530.0 

535.0 

540.1 

545.2 

550.3 

555.4 

560.5 

565.6 

570.8 

16 

576.0 

581.2 

586.4 

591.7 

597.0 

602.3 

607.6 

612.9 

618.2 

623.6 

17 

629.0 

634.4 

639.8 

645.3 

650.8 

056.3 

661.8 

667.3 

672.8 

678.4 

18 

681.0 

689.6 

695.2 

700.9 

706.6 

712.3 

718.0 

723.7 

729.4 

735.2 

19 

741.0 

746.8 

752.6 

758.5 

764.4 

770.3 

776.2 

782.1 

788.0 

794.0 

20 

800.0 

806.0 

812.0 

818.1 

824.2 

830.3 

836.4 

842.5 

848.6 

854.8 

21 

861.0 

867.2 

873.4 

879.7 

886.0 

892.3 

898.6 

904.9 

911.2 

917.6 

22 

924.0 

930.4 

936.8 

943.3 

949.8 

956.3 

902.8 

969.3 

975.8 

982.4 

23 

989.0 

995.6 

1002.2 

1008.9 

1015.6 

1022.3 

1029.0 

1035.7 

1042.4  1049.2 

24 

1056.0 

1062.8 

1069.6 

1076.5 

1083.4 

1090.3 

1097.2 

1104.1 

1111.0  1118.0 

25 

1125.0 

1132.0 

1139.0 

1146.1 

1153.2 

1160.3  1167.4 

1174.5  1181.6  1188.8 

27 

1196.0 

1203.2 

1210.4 

1217.7 

1225.0 

1232.3 

1239.6 

1246.9  1254.2 

1261.6 

26 

1269.0 

1276.4 

1283.8 

1291.3 

1298.8 

1306.3 

1313.8 

1321.3 

1328.8  1.336.4 

28 

1344.0 

1351.6 

1.359.2 

1366.9 

1374.6 

1382.3  1390.0 

1397.7 

1405.4  1413.2 

29 

1421.0 

1428.8 

1436.6 

1444.5 

1452.4 

1460.3  1468.2 

1476  1 

1484.0  1492.0 

Base,  26 

=  30  feet.    Side  slopes  1  to  1. 

C.  H. 

.0 

.1 

2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

3.0 

6.0 

9.1 

12.2 

15.3 

18.4 

21.5 

24.6 

27.8 

1 

31.0 

34.2 

37.4 

40.7 

44.0 

47.3 

50.6 

53.9 

57.2 

60.6 

2 

64.0 

67.4 

70.8 

74.3 

77.8 

81.3 

84.8 

88.3 

91.8 

95.4 

3 

99.0 

102.6 

106.2 

109.9 

113.6 

117.3 

121.0 

124.7 

128.4 

132.2 

4 

136.0 

139.8 

143.6 

147.5 

151.4 

155.3 

159.2 

163.1 

167.0 

171.0 

5 

175.0 

179.0 

183.0 

187.1 

191.2 

195.3 

199.4 

203.5 

207.6 

211.8 

6 

216.0 

220.2 

224.4 

228.7 

233.0 

237.3 

241.6 

245.9 

250.2 

254.6 

7 

259.0 

263.4 

267.8 

272.3 

276.8 

281.3 

285.8 

290.3 

294.8 

299.4 

8 

304.0 

308.6 

313.2 

317.9 

.322.6 

327.3 

332.0 

330. 7 

341.4 

346.2 

9 

351.0 

355.8 

360.6 

365.5 

370.4 

375.3 

380.3 

385.1 

390.0 

395,0 

10 

400.0 

405.0 

410.0 

415.1 

420.2 

425.3 

430.4 

435.5 

440.6 

445,8 

11 

451.0 

456.2 

401.4 

466.7 

472.0 

477.3 

482.6 

487.9 

493.2 

498,6 

12 

504.0 

509.4 

514.8 

520.3 

.525.8 

531.3 

.536.8 

.542,3 

547.8 

553,4 

13 

559.0 

564.6 

570.2 

575.9 

581.6 

587.3 

593.0 

598.7 

604.4 

610.2 

14 

616.0 

621.8 

627.6 

633.5 

639.4 

645.3 

651.2 

057.1 

663.0 

669.0 

15 

675.0 

681.0 

687.0 

693.1 

699.2 

705.3 

711.4 

717.5 

723.6 

729.8 

16 

736.0 

742.2 

748.4 

754.7 

761.0 

767.3 

773.6 

779.9 

786.2 

792.6 

17 

799.0 

805.4 

811.8 

818.3 

824.8 

831.3 

837.8 

844.3 

850.8 

857.4 

18 

864.0 

870.6 

877.2 

883.9 

890.6 

897.3 

904.0 

910.7 

917.4 

924.2 

19 

931.0 

937.8 

944.6 

951.5 

958.4 

965.3 

972.2 

979.1 

986.0 

993.0 

20 

1000.0 

1007.0 

1014.0 

1021.1 

1028.2 

1035.3 

1042.4 

1049.5  1056.6  1063.8 

21 

1071.0 

1078.2 

1085.4 

1092.7 

1100.0 

1107.3 

1114.6 

1121.9  1129.2  1136.6 

22 

1144.0 

1151.4 

1158.8 

1166.3 

1173.8 

1181.3 

1188.8 

1196.3 

1203.8 

1211.4 

23 

1219.0 

1226.6 

1234.2 

1241.9 

1249.6 

1257.3  1265.0 

1272.7 

1280.4 

1288.2 

24 

1296.0 

1303.8 

1311.6 

1319.5 

1327.4 

1335.3  1343.2 

1351.1 

1359.0 

1367.0 

25 

1375.0 

1383.0 

1391.0 

1399.1 

1407.2 

1415.3  1423.4 

1431.5 

1439.6 

1147.8 

26 

1456.0 

1464.2 

1472.4 

1480.7 

1489.0 

1497.3 

1.505.6 

1513.9 

1522.2 

1530.6 

27 

1539.0 

1547.4 

1555.8 

1564.3 

1572.8 

1581.3 

1589.8 

1598.3 

1006.8 

1015.4 

28 

1624.0 

1632.6 

1641.2 

1649.9 

1658.6 

1667.3  1676.0  1684.7 

1693.4 

1702.2 

29 

1711.0 

1719.8 

1728.6 

1737.5 

1746.4 

17.55.3 

1764.2 

1773.1 

1782.0 

1791.0 

374:       TABLE   XVII.— AREAS   OF  LEVEL   SECTIONS. 

Base,  26  =  16  feet.    Side  slopes  J  to  1. 


C.  H. 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.7 

.8 

.9 

0 

0.0 

1.6 

3.2 

4.8 

6.4 

8.1 

9.7 

11.3 

13.0 

14.6 

1 

16.3 

17.9 

19.6 

21.2 

22.9 

24.6 

26.2 

27.9 

29.6 

31.3 

2 

33.0 

34.7 

36.4 

38.1 

39.8 

41.6 

43.3 

45.0 

46.8 

48.5 

3 

50.3 

52.0 

.53.8 

55.5 

57.3 

59.1 

60.8 

62.6 

64.4 

66.2 

4 

68.0 

69.8 

71.6 

73. '4 

75.2 

77.1 

78.9 

80.7 

82.6 

84.4 

o 

86.3 

88.1 

90.0 

91.8 

93.7 

95.6 

97.4 

99.3 

101.2 

103.1 

6 

105.0 

106.9 

108.8 

110.7 

112.6 

114.6 

116.5 

118.4 

120.4 

122.3 

7 

124.3 

126.2 

128.2 

130.1 

132.1 

134.1 

136.0 

138.0 

140.0 

142.0 

8 

144.0 

146.0 

148.0 

150.0 

152.0 

154.1 

156.1 

158.1 

160.2 

162.2 

9 

164.3 

166.3 

168.4 

170.4 

172.5 

174.6 

176.6 

178.7 

180.8 

182.9 

10 

185.0 

187.1 

189.2 

191.3 

193.4 

195.6 

197.7 

199,8 

202.0 

204.1 

11 

206.3 

208.4 

210.6 

212.7 

214.9 

217.1 

219.2 

221.4 

223.6 

225.8 

12 

228.0 

230.2 

232.4 

234.6 

236.8 

239.1 

241.3 

243.5 

245.8 

248.0 

13 

250.3 

252.5 

254.8 

257.0 

259.3 

261.6 

263.8 

266.1 

268.4 

270.7 

14 

273.0 

275.3 

277.6 

279.9 

282.2 

284.6 

286.9 

289.2 

291.6 

293.9 

16 

296.3 

298.6 

301.0 

303.3 

305.7 

308.1 

310.4 

312.8 

315.2 

317.6 

16 

320.0 

322.4 

324.8 

327.2 

329.6 

332,1 

334.5 

336.9 

339.4 

341.8 

17 

344.3 

346.7 

349.2 

351.6 

354.1 

356.6 

359.0 

361.5 

364.0 

366.5 

18 

369.0 

371.5 

374.0 

376.5 

379.0 

381.6 

384.1 

386.6 

389.2 

391.7 

19 

394.3 

396.8 

399.4 

401.9 

404.5 

407.1 

409.6 

412.2 

414.8 

417.4 

20 

420.0 

422.6 

425.2 

427.8 

430.4 

433.1 

435.7 

438.3 

441.0 

443.6 

21 

446.3 

448.9 

451.6 

454.2 

456.9 

459.6 

462.2 

464.9 

467.6 

470.3 

22 

473.0 

475.7 

478.4 

481.1 

483.8 

486.6 

489.3 

492.0 

494.8 

497.5 

23 

500.3 

503.0 

505.8 

508.5 

511.3 

514.1 

516.8 

519.6 

522.4 

525.2 

24 

528.0 

530.8 

533.6 

536.4 

539.2 

542.1 

544.9 

547.7 

550.6 

553.4 

25 

556.3 

559.1 

562.0 

564.8 

567.7 

570.6 

573.4 

576.3 

579.2 

582.1 

26 

585.0 

587.9 

590.8 

593.7 

596.6 

599.6 

602.5 

605.4 

608.4 

611.3 

27 

614.3 

617.2 

620.2 

623.1 

626.1 

629.1 

632.0 

635.0 

638.0 

641.0 

28 

644.0 

647.0 

650.0 

653.0 

656.0 

659.1 

662.1 

665.1 

668.2 

671.2 

29 

674.3 

677.3 

680.4 

683.4 

686.5 

689.6 

692.6 

695.7 

698.8 

701.9 

Base,  26 

=  18  feet.    Side  slopes  i  to  1. 

C.  H. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

1.8 

3.6 

5.4 

7.2 

9.1 

10.9 

12.7 

14.6 

16.4 

1 

18.3 

20.1 

22.0 

23.8 

25.7 

27.6 

29.4 

31.3 

33.2 

35.1 

2 

37.0 

38.9 

40.8 

42.7 

44.6 

46.6 

48.5 

50.4 

52.4 

54.3 

3 

56.3 

58.2 

60.2 

62.1 

64.1 

66.1 

68.0 

70.0 

.0 

74.0 

4 

76.0 

78.0 

80.0 

82.0 

84.0 

86.1 

88.1 

90.1 

o 

94.2 

o 

96.3 

98.3 

100.4 

102.4 

104.5 

106.6 

108.6 

110.7 

112.8 

114.9 

6 

117.0 

119.1 

121.2 

123.3 

125.4 

127.6 

129.7 

131.8 

134.0 

136.1 

7 

138.3 

140.4 

142.6 

144.7 

146.9 

149.1 

151.2 

153.4 

155.6 

157.8 

8 

1«0.0 

162.2 

164.4 

166.6 

168.8 

171.1 

173.3 

175.5 

177.8 

180.0 

9 

182.3 

184.5 

186.8 

189.0 

191.3 

193.6 

195.8 

198.1 

200.4 

202.7 

10 

205.0 

207.3 

209.6 

211.9 

214.2 

216.6 

218.9 

221.2 

223.6 

225.9 

11 

228.3 

230.6 

233.0 

235.3 

237.7 

240.1 

242.4 

244.8 

247.2 

249.6 

12 

252.0 

254.4 

256.8 

259.2 

261.6 

264.1 

266.5 

268.9 

271.4 

273.8 

13 

276.3 

278.7 

281.2 

283.6 

286.1 

288.6 

291.0 

293.5 

296.0 

298.5 

14 

301.0 

303.5 

306.0 

308.5 

311.0 

313.6 

316.1 

318.6 

321.2 

323.7 

15 

336.3 

328.8 

331.4 

333.9 

336.5 

339.1 

341.6 

344.2 

346.8 

349.4 

16 

352.0 

354.6 

357.2 

359.8 

362.4 

365.1 

367.7 

370.3 

373.0 

375.6 

17 

378.3 

380.9 

383.6 

386.2 

388.9 

391.6 

394.2 

396.9 

399.6 

402.3 

18 

405.0 

407.7 

410.4 

413.1 

415.8 

418.6 

421.3 

424.0 

426.8 

429.5 

19 

432.3 

435.0 

437.8 

440.5 

443.3 

446.1 

448.8 

451.6 

454.4 

457.2 

20 

460.0 

462.8 

465.6 

468.4 

471.2 

474.1 

476.9 

479.7 

482.6 

485.4 

21 

488.3 

491.1 

494.0 

496.8 

499.7 

502.6 

505.4 

508.3 

511.2 

514.1 

22 

517.0 

519.9 

522.8 

525.7 

528.6 

531.6 

534.5 

537.4 

540.4 

543.3 

23 

546.3 

549.2 

552.2 

555.1 

558.1 

561.1 

564.0 

567.0 

570.0 

573.0 

24 

576.0 

579.0 

582.0 

585.0 

588.0 

591.1 

594.1 

597.1 

600.2 

603.2 

25 

606.3 

609.3 

612.4 

615.4 

618.5 

621.6 

624.6 

627.7 

630.8 

633.9 

26 

637.0 

640.1 

643.2 

646.3 

649.4 

652.6 

655.7 

658.8 

662.0 

665.1 

27 

668.3 

671.4 

674.6 

677.7 

6S0.9 

684.1 

687.2 

690.4 

693.6 

696.8 

28 

700.0 

703.2 

706.4 

709.6 

712.8 

716.1 

719.3 

722.5 

725.8 

729.0 

29 

732.3 

735.5 

738.8 

742.0 

745.3 

748.6 

751.8 

755.1 

758.4 

761.7    i 

XVIII.-AREA  CORRECTIONS  FOR  THREE-LEVEL  GROUND. 
Correction  —  (/i,n  —  ho)^s.    (See  331.) 

SIDE  SLOPES   U   TO   1. 


375 


hm-ho 
0 

.0 

.1 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0.0 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

1.2 

1 

1.5 

1.8 

2.2 

2.5 

2,9 

3.4 

3.8 

4.3 

4  9 

5.4 

2 

6.0 

6.6 

7.3 

7.9 

8.6 

9.4 

10.1 

10.9 

11.8 

12.6 

3 

13.5 

14.4 

15.4 

16.3 

17.3 

18.4 

19.4 

20.5 

21.7 

22.8 

4 

24.0 

25.2 

26.5 

27.7 

29.0 

30.4 

31.7 

33.1 

34.6 

36.0 

5 

37.5 

39.0 

40.6 

42.1 

43,7 

45.4 

47.0 

48.7 

50.5 

5  J  2 

0 

54.0 

55.8 

57.7 

59.5 

61.4 

63.4 

65.3 

67.3 

69.4 

71.4 

7 

73.5 

75.6 

77.8 

79.9 

82.1 

84.4 

86.6 

88.9 

91.3 

93  6 

8 

96.0 

98.4 

100.9 

103.3 

105.8 

108.4 

110.9 

113.5 

116.2 

118.8 

9 

121.5 

124.2 

127.0 

129.7 

13-.'.5 

135.4 

138.2 

141.1 

144.1 

147.0 

10 

150.0 

153.0 

156.1 

159.1 

162.2 

165.4 

168.5 

171.7 

175.0 

178.2 

11 

181.5 

184.8 

188.2 

191.5 

194.9 

198.4 

201.8 

205.3 

208.9 

212.4 

SIDE   SLOPES 

1   TO   1. 

hm-ho 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.7 

.8 

.9 

0 

0.0 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

1 

1.0 

1.2 

1.4 

1.7 

2.0 

2.3 

2.6 

2.9 

3.2 

3.6 

2 

4.0 

4.4 

4.8 

5.3 

5.8 

6.3 

6.8 

7.3 

7.8 

8.4 

3 

9.0 

9.6 

10.2 

10.9 

11.6 

12.3 

13.0 

13.7 

14.4 

15.2 

4 

16.0 

16.8 

17.6 

18.5 

19.4 

20.3 

21.2 

22.1 

23.0 

24.0 

5 

25.0 

26.0 

27.0 

28.1 

29.2 

30.3 

31.4 

32.5 

33.6 

34.8 

6 

36.0 

37.2 

38.4 

39.7 

41.0 

42.3 

43.6 

44.9 

46.2 

47.6 

7 

49.0 

50.4 

51.8 

53.3 

54.8 

56.3 

57.8 

59.3 

60.8 

62.4 

8 

64.0 

65.6 

67.2 

68.9 

70.6 

72.3 

74.0 

75.7 

77.4 

79.2 

9 

81.0 

82.8 

84.6 

86.5 

88.4 

90.3 

92.2 

94.1 

96.0 

98.0 

10 

100.0 

102.0 

104.0 

106.1 

108.2 

110.3 

112.4 

114.5 

116.6 

118.8 

11 

121.0 

123.2 

125.4 

127.7 

130.0 

132.3 

134.6 

136.9 

139.2 

141.6 

SIDE  SLOPES  i 

TO   1, 

hm-ho 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

1 

0.5 

0.6 

0  7 

0.8 

1.0 

1.1 

1.3 

1.4 

1.6 

1.8 

2 

2.0 

2.2 

2.4 

2.6 

2.9 

3.1 

3.4 

3.6 

3.9 

4.2 

3 

4.5 

4.8 

5.1 

5.4 

5.8 

6.1 

6.5 

6.8 

7.2 

T.6 

4 

8.0 

8.4 

8.8 

9.2 

9.7 

10.1 

10.6 

11  0 

11.5 

12.0 

5 

12.5 

13.0 

13.5 

14.0 

14.6 

15.1 

15.7 

16.2 

16.8 

17.4 

6 

18.0 

18.6 

19.2 

19.8 

20.5 

21.1 

21.8 

22.4 

23.1 

23.8 

7 

24.5 

25.2 

25.9 

26.6 

27.4 

28.1 

28.9 

29.6 

30.4 

31.2 

8 

32.0 

32.8 

33.6 

34.4 

35.3 

36.1 

37.0 

37.8 

38.7 

39.6 

9 

40.5 

41.4 

42.3 

43.2 

44.2 

45.1 

46.1 

47.0 

48.0 

49.0 

10 

50.0 

51.0 

52.0 

53.0 

54.1 

55.1 

56.2 

57.2 

58.3 

59.4 

11 

60.5 

61.6 

62.7 

63.8 

65.0 

66.1 

67.3 

68.4 

69.6 

70.8 

SIDE  SLOPES   J 

TO  1. 

hm-ho 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

1 

0.3 

0.3 

0.4 

0.4 

0.5 

0.6 

0.6 

0.7 

0.8 

0.9 

2 

1.0 

1.1 

1.2 

1.3 

1.4 

1.6 

1.7 

1.8 

2.0 

2.1 

3 

2.3 

2.4 

2.6 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.8 

4 

4  0 

4.2 

4.4 

4.6 

4.8 

5.1 

5.3 

5.5 

5.8 

6.0 

5 

6  3 

6.5 

6.8 

7.0 

7.3 

7.6 

7.8 

8.1 

8.4 

8.7 

6 

9.0 

9  3 

9.6 

9.9 

10.2 

10.6 

10.9 

11.2 

11.6 

11.9 

7 

12.3 

12.6 

13.0 

13.3 

13.7 

14.1 

14.4 

14.8 

15.2 

15.6 

8 

16.0 

16.4 

16.8 

17.2 

17  6 

18.1 

18.5 

18.9 

19.4 

19.8 

9 

20.3 

20.7 

21.2 

21.6 

22.1 

22.6 

23.0 

23.5 

24.0 

24.5 

10 

25.0 

25.5 

26.0 

26.5 

27.0 

27.6 

28.1 

28.6 

29.2 

29.7 

11 

30.3 

30.8 

21.4 

31.9 

32.5 

33.1 

33.6 

34.2 

34.8 

35.4 

376  XIX.— CUBIC  YARDS  PER  100  FEET. 


SLOPES  i 


1. 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

22 

24 

26 

28 

1 

45 

53 

60 

68 

82 

90 

97 

105 

2 

93 

107 

122 

137 

167 

181 

196 

211 

3 

142 

163 

186 

208 

253 

275 

297 

319 

4 

193 

222 

252 

281 

341 

370 

400 

430 

5 

245 

282 

319 

356 

431 

468 

505 

542 

6 

300 

344 

389 

433 

522 

567 

611 

656 

7 

356 

408 

460 

512 

616 

668 

719 

771 

8 

415 

474 

533 

593 

711 

770 

830 

889 

9 

475 

542 

808 

675 

808 

875 

942 

1008 

10 

537 

611 

685 

759 

907 

981 

1056 

1130 

11 

601 

682 

764 

845 

1008 

1090 

1171 

1353 

12 

667 

756 

&44 

933 

1111 

1200 

1289 

1378 

13 

734 

831 

926 

102:3 

1216 

1312 

1408 

1505 

14 

804 

907 

1010 

1115 

1322 

1426 

1530 

1633 

15 

8:5 

986 

1096 

1208 

1431 

1542 

1653 

1764 

16 

948 

1067 

11!^ 

1304 

1541 

1659 

1778 

1896 

17 

1023 

1149 

1274 

1401 

1653 

1779 

1905 

2031 

18 

1100 

1233 

1366 

1500 

1767 

1900 

2033 

2167 

19 

1179 

1319 

1460 

1601 

1882 

2023 

21&4 

2305 

20 

1259 

1407 

1555 

1704 

2000 

2148 

2296 

2444 

21 

1342 

1497 

1653 

1808 

2119 

2275 

2431 

2586 

22 

1426 

1589 

1752 

1915 

2241 

2404 

2567 

2730 

23 

1512 

1682 

ia53 

2023 

2364 

2534 

2705 

2875 

24 

1600 

1778 

1955 

2ia3 

2489 

2667 

2*44 

3022 

25 

1690 

1875 

2060 

2245 

2616 

;2801 

2986 

3171 

26 

1781 

1974 

2166 

2359 

2744 

2937 

3130 

3322 

27 

1875 

2075 

2274 

2475 

2875 

3075 

3275 

3475 

28 

1970 

2178 

2384 

2593 

3007 

3215 

3422 

3630 

29 

2068 

2282 

2496 

2712 

3142 

3358 

3571 

3786 

30 

2167 

2389 

2610 

2833 

3278 

3500 

3722 

3944 

31 

2268 

2497 

2726 

2956 

3416 

3645 

3875 

4105 

32 

2370 

2607 

2*44 

3081 

3556 

3793 

4030 

4267 

33 

2475 

2719 

2964 

3208 

3697 

3942 

4186 

4431 

34 

2581 

283:B 

3085 

3337 

3841 

4093 

4344 

4596 

35 

2690 

2949 

3208 

3468 

3986 

-4245 

4505 

4764 

36 

2800 

3067 

3333 

3600 

41:33 

4400 

4667 

4933 

37 

2912 

3186 

3460 

3734 

4282 

4556 

4831 

5105 

38 

3026 

3307 

3589 

3870 

4433 

4715 

4996 

5278 

39 

3142 

3431 

3719 

4008 

4586 

4875 

5164 

5453 

40 

3259 

3556 

3852 

4148 

4741 

5037 

5333 

5630 

41 

3379 

3682 

3986 

4290 

4897 

5201 

5505 

5808 

42 

3500 

3811 

4122 

4433 

5056 

5367 

5678 

5989 

43 

3623 

3942 

4260 

4579 

5216 

5534 

5853 

6171 

44 

3748 

4074 

4400 

4726 

5378 

5704 

6030 

6356 

45 

3875 

4208 

4541 

4875 

5542 

5875 

6208 

6542 

46 

4004 

4*44 

4684 

5026 

5707 

6048 

6389 

6730 

47 

4134 

4482 

48:30 

5179 

5875 

6223 

6571 

6919 

48 

4267 

4622 

4978 

5333  1 

6044 

6400 

6756 

7111 

49 

4401 

4764 

5127 

5490 

6216 

6579 

6942 

7305 

50 

4537 

4907 

5278 

5648 

6389 

j 

6759 

7130 

7500 

51 

4675 

5053 

5430 

5808  ' 

6564 

6942 

7319 

7697 

52 

4815 

5200 

5584 

5970 

6741 

7126 

7511 

7896 

53 

4956 

5349 

5741 

6134 

6919 

7312 

7705 

8097 

54 

5100 

5500 

5900 

6300 

rioo 

7500 

7900 

«:300 

55 

5245 

565:3 

6060 

6468 

7282 

7690 

8097 

8505 

56 

5393 

5807 

6222 

6637  i 

7467  : 

7881 

8296  1 

8711 

57 

5542 

5964 

6386 

6808  1 

7653  ; 

8075 

*497 

8919 

58 

569:3 

6122 

6552 

6981 

7*41  ' 

8270 

8700  : 

9130 

59 

5845 

6282 

6719  1 

7156 

8031  ; 

8468 

8905 

9342 

60 

6000 

&444 

6889 

7.3,33 

8222 

8667 

9111 

9556 

XIX.— CUBIC  YARDS  PER  100  FEET.     SLOPES  ^  :  1.  B?'? 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

22 

24 

26 

28 

1 

46 

54 

61 

69 

83 

91 

98 

106 

2 

96 

111 

126 

141 

170 

185 

200 

215 

3 

150 

172 

194 

217 

261 

283 

306 

328 

4 

207 

237 

267 

296 

356 

385 

415 

444 

5 

269 

306 

343 

380 

4.54 

491 

528 

565 

6 

333 

378 

422 

467 

556 

600 

644 

689 

r 

402 

454 

506 

557 

6G1 

713 

765 

817 

8 

474 

533 

593 

652 

770 

830 

889 

948 

9 

550 

617 

683 

750 

883 

950 

1017 

1083 

10 

630 

704 

778 

852 

1000 

1074 

1148 

1222 

11 

713 

794 

876 

957 

1120 

1202 

1283 

1365 

12 

800 

889 

978 

1067 

1244 

1333 

1422 

1511 

13 

891 

987 

1083 

1180 

1372 

1469 

1565 

1661 

14 

985 

1089 

1193 

1296 

1504 

1607 

1711 

1815 

15 

1083 

1194 

1306 

1417 

1639 

1750 

1861 

1972 

16 

1185 

1304 

1422 

1541 

1779 

1896 

2015 

2133 

17 

1291 

1417 

1543 

1669 

1920 

2046 

2172 

2298 

18 

1400 

1533 

1667 

1800 

2067 

2200 

2333 

2467 

19 

1513 

1654 

1794 

1935 

2217 

2357 

2498 

2639 

20 

1630 

1778 

1926 

2074 

2370 

2519 

2667 

2815 

21 

1750 

1906 

2061 

2217 

2528 

2683 

2839 

2994 

22 

1874 

2037 

2200 

2363 

2689 

2852 

3015 

3178 

23 

2002 

2172 

2343 

2513 

2854 

3024 

3194 

3365 

24 

2133 

2311 

2489 

2667 

3022 

3200 

3378 

3556 

25 

2269 

2454 

2639 

2824 

3194 

3380 

3565 

3750 

26 

2407 

2600 

2793 

2985 

3370 

35C3 

3756 

3948 

27 

2550 

2750 

2950 

3150 

3550 

3750 

3950 

4151 

28 

2696 

2904 

3111 

3319 

3733 

3941 

4148 

4356 

29 

2846 

3061 

3276 

3491 

3920 

4135 

4350 

4565 

30 

3000 

3222 

3444 

3667 

4111 

4333 

4556 

4778 

31 

3157 

3387 

3617 

3846 

4306 

4535 

4765 

4994 

32 

3319 

3556 

3793 

4030 

4504 

4741 

4978 

5215 

33 

3483 

3728 

3972 

4217 

4706 

4950 

5194 

5439 

34 

3652 

3904 

4156 

4407 

4911 

5163 

5415 

5667 

35 

3824 

4083 

4343 

4602 

5120 

5380 

5639 

5898 

36 

4000 

4267 

4533 

4800 

5333 

5600 

5867 

6133 

37 

4180 

4454 

4728 

5002 

5550 

5824 

6098 

6372 

38 

4363 

4644 

4926 

5207 

5770 

6052 

6333 

6615 

39 

4550 

4839 

5128 

5417 

5994 

6283 

6572 

6861 

40 

4741 

5037 

5333 

5630 

6222 

6519 

6815 

7111 

41 

4935 

5239 

5543 

5846 

6454 

6757 

7061 

7365 

42 

5133 

5444 

5756 

6067 

6689 

7000 

7311 

7622 

43 

5335 

5654 

5972 

6291 

6928 

7246 

7565 

7883 

44 

5541 

5867 

6193 

6519 

7170 

7496 

7822 

8148 

45 

5750 

6083 

6417 

6750 

7417 

7750 

8083 

8417 

46 

5963 

6304 

6644 

6985 

7667 

8007 

8348 

8689 

47 

6180 

6528 

6876 

7224 

7920 

8269 

8617 

8965 

48 

6400 

6756 

7111 

7467 

8178 

8533 

8889 

9244 

49 

6624 

6987 

7350 

7713 

8439 

8802 

9165 

9528 

50 

6852 

7222 

7593 

7963 

87G4 

9074 

9444 

9815 

51 

7083 

7461 

7839 

8217 

8972 

9350 

9728 

10106 

52 

7319 

7704 

8089 

8474 

9244 

9630 

10015 

10400 

53 

7557 

7950 

8343 

8735 

9520 

9913 

10306 

10698 

54 

7800 

8200 

8000 

9000 

9800 

10200 

10600 

11000 

55 

8046 

8454 

8S61 

9269 

10083 

10491 

10898 

11306 

56 

8296 

8711 

9126 

9541 

10370 

10785 

11200 

11615 

57 

8550 

8972 

9394 

9817 

10661 

11083 

11506 

11928 

58 

8807 

9237 

9667 

10096 

10956 

11385 

11815 

12244 

59 

9069 

9506 

9943 

10380 

11254 

11691 

12128 

12565 

60 
t^ 

9333 

9778 

10222 

10667 

11556 

12000 

12444 

12889 
1 

378  XIX. -CUBIC  YARDS  PER  100  FEET.     SLOPES  1  :  1. 


Depth 

Base 

Base 

Base 

12 

14 

16 

1 

48 

56 

G3 

2 

104 

119 

133 

3 

167 

189 

211 

4 

237 

267 

296 

5 

315 

352 

389 

6 

400 

444 

489 

7 

493 

544 

596 

8 

593 

652 

711 

9 

700 

767 

833 

10 

815 

889 

963 

11 

937 

1019 

1100 

12 

10G7 

1156 

1244 

13 

1204 

1300 

1396 

14 

1348 

1452 

1556 

15 

1500 

1611 

1722 

16 

1659 

1778 

1896 

17 

1826 

1952 

2078 

18 

2000 

2i;33 

2207 

19 

2181 

2322 

2463 

20 

2370 

2519 

2667 

21 

2567 

27.22 

2878 

22 

2770 

2933 

3096 

23 

2981 

3152 

3322 

24 

3200 

3378 

3556 

25 

3426 

3611 

3796 

26 

3659 

3852 

4044 

27 

3900 

4100 

4300 

28 

4148 

4356 

4503 

29 

4404 

4019 

4833 

30 

4667 

4889 

5111 

31 

4937 

5167 

5396 

32 

5215 

5452 

5089 

33 

5500 

5744 

5989 

34 

5793 

6044 

6296 

35 

6093 

6352 

6611 

36 

6400 

6667 

6933 

37 

6715 

6989 

7263 

38 

7037 

7319 

7600 

39 

7367 

7656 

7944 

40 

7704 

8000 

8296 

41 

8048 

8352 

8656 

42 

8400 

8711 

9022 

43 

8759 

9078 

9396 

44 

9126 

9452 

9778 

43 

9500 

9833 

10167 

46 

9881 

10222 

10563 

47 

10270 

10619 

10967 

48 

10667 

11022 

11378 

49 

11070 

114a3 

11796 

50 

11481 

11852 

12222 

51 

11900 

12278 

12656 

52 

12326 

12711 

13096 

53 

12759 

131.52 

1:3544 

54 

13200 

13G00 

14000 

55 

13648 

14056 

14463 

56 

14104 

14519 

14933 

57 

14567 

14989 

15411 

58 

15037 

15467 

15896 

59 

15515 

159.52 

16389 

60 

16000 

1&444 

16889 

Base 
18 

70 
148 
233 
326 
426^ 
533 
648 
770 
900 
1037 

1181 
13:3:3 
1493 
1659 
183:3 
2015 
2204 
2400 
2604 
2815 

3033 
;3259 
^493 
37^3 
3981 
4237 
4500 
4770 
5048 
5333 

5626 
592G 
62:33 
6548 
6870 
7200 
7537 
7881 
82:33 
8593 

8959 
9333 
9715 
10104 
10500 
10904 
11315 
11733 
121.59 
12593 

13033 
i:3481 
139:37 
14400 
14870 
1.5348 
15833 
16326 
16826 
17333 


Base 
20 


Base 
28 


Base 
30 


Base 
32 


78 
163 
256 
356 
463 
578 
700 
830 
967 
1111 

1263 
1422 
1589 
1763 
1944 
2133 
2330 
2533 
2744 
2963 

3189 
3422 
3663 
3911 
4167 
4430 
4700 
4978 
5263 
5556 

5856 
6163 
6478 
6800 
7130 
7467 
7811 
8163 
8522 
8889 

9263 
9644 
10033 
10430 
10833 
11244 
11663 
12089 
12522 
12963 

13411 
1:3867 
14330 
14800 
15278 
15763 
16256 
16756 


107 

222 

344 

474 

611 

756 

907 

1067 

1233 

1407 

1589 
1778 
1974 
2178 
2389 
2607 
2833 
3067 
3307 
3556 

3811 
4074 
4:344 
4022 
4907 
5200 
5500 
5807 
6122 
6444 

6774 
7111 
7456 

7807 
8167 
8533 
8907 
9289 
9678 
10074 

10478 
10889 
11:307 
11733 
12167 
12607 
13056 
13511 
13974 
14444 

14922 
15407 
15900 
16400 
16907 
17422 
17944 
18474 
19011 
19556 


115 

237 

367 

504 

648 

800 

959 

1126 

1:300 

1481 

1670 
1867 
2070 
2281 
2500 
2726 
2959 
3200 
3448 
3704 

3967 
4237 
4515 
4800 
5093 
5393 
5700 
6015 
6337 
6667 

7004 
7348 
7700 
8059 
8426 
8800 
9181 
9570 
9967 
10370 

10781 
11200 
11626 
12059 
12.500 
12948 
13404 
13867 
14337 
14815 

15300 
15793 
16293 
16800 
17315 
17837 
18367 
18904 
19448 
20000 


122 

252 

389 

533 

685 

844 

1011 

1185 

1367 

1556 

1752 
1956 
2167 
2385 
2611 
2844 
3085 
3:333 
35S9 
3852 

4122 
4444 

46S5 
4978 
5278 
5585 
5900 
6222 
6552 
6889 

723:3 
7585 
7944 
8311 
8685 
9067 
9456 
9852 
10256 
10667 

11085 
11511 
11944 
123S5 
128:33 
13289 
13752 
14222 
14700 
15185 

1.5678 
16178 
16685 
17200 
17722 
18252 
18789 
19333 
10885 
20444 


XIX.— CUBIC  YARDS  PER  100  FEET.     SLOPES  U  :  1-  ^'^^ 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

20 

28 

30 

32 

1 

50 

57 

65 

72 

80 

109 

117 

124 

2 

111 

126 

141 

156 

170 

230 

244 

259 

3 

183 

206 

228 

250 

272 

361 

383 

406 

4 

267 

296 

326 

356 

385 

504 

533 

563 

5 

361 

398 

435 

472 

509 

657 

694 

731 

6 

467 

511 

556 

600 

644 

822 

867 

911 

7 

583 

635 

687 

739 

791 

998 

1050 

1102 

8 

711 

770 

830 

889 

948 

1185 

1244 

1304 

9 

850 

917 

983 

1050 

1116 

1383 

1450 

1517 

10 

1000 

1074 

1148 

1222 

1296 

1593 

1667 

1741 

11 

1161 

1243 

1324 

1406 

1487 

1813 

1894 

1976 

12 

1333 

1422 

1511 

1600 

1689 

2044 

2133 

2222 

13 

1517 

1613 

1709 

1806 

1902 

2287 

2383 

2480 

14 

1711 

1815 

1919 

2022 

2126 

2541 

2644 

2748 

15 

1917 

2028 

2139 

2250 

2361 

2806 

2917 

3028 

16 

2133 

2252 

2370 

2489 

2607 

3081 

3200 

a319 

17 

2361 

2487 

2613 

2739 

2865 

3369 

3494 

3620 

18 

2600 

2733 

2867 

3000 

3133 

3667  ■ 

3800 

3933 

19 

2850 

2991 

3131 

3272 

3413 

3976 

4117 

4257 

20 

3111 

3259 

3407 

3556 

3704 

4296 

4444 

4592 

21 

3383 

3539 

3694 

3850 

4005 

4628 

4783 

4939 

3667 

3830 

3993 

4156 

4318 

4970 

5133 

5296 

23 

3961 

4131 

4302 

4472 

4642 

5324 

5494 

5665 

24 

4267 

4444 

4622 

4800 

4978 

5689 

5867 

6044 

25 

4583 

4769 

4954 

5139 

5324 

6065 

6250 

6435 

2« 

4911 

5104 

5296 

5489 

5681 

6452 

6644 

6837 

2? 

5250 

5450 

5650 

5850 

6050 

6850 

7050 

7250 

28 

5600 

5807 

6015 

6222 

6430 

7259 

7467 

7674 

29 

5961 

6176 

6391 

6606 

6820 

7680 

7894 

8109 

30 

6333 

6556 

6778 

7000 

7222 

8111 

8333 

8555 

31 

6717 

6946 

7176 

7406 

7635 

8554 

8783 

9013 

3C' 

7111 

7348 

7585 

7822 

8059 

9007 

9244 

9482 

33 

7517 

7761 

8006 

8250 

8494 

9472 

9717 

9962 

34 

7933 

8185 

8437 

8689 

8941 

9948 

10200 

10452 

35 

8361 

8620 

8880 

9139 

9398 

10435 

10694 

10954 

36 

8800 

9067 

9333 

9600 

9867 

10933 

11200 

11467 

37 

9^50 

9524 

9798 

10072 

10346 

11443 

11717 

11991 

38 

9711 

9993 

10274 

10556 

10.S3? 

11963 

12244 

12526 

39 

10183 

10472 

10761 

11050 

11339 

12494 

12783 

13072 

40 

10667 

10963 

11259 

11556 

11852 

13037 

13333 

13630 

41 

11161 

11465 

11769 

12072 

12376 

13591 

13894 

14198 

42 

11667 

11978 

12289 

12600 

12911 

14156 

14467 

14778 

43 

12183 

12502 

12820 

13139 

13457 

14731 

15050 

15369 

44 

12711 

13037 

13303 

13689 

14015 

15319 

15644 

15970 

45 

13250 

13583 

13917 

14250 

14583 

15917 

16250 

16583 

46 

13800 

14141 

14481 

14822 

15163 

16526 

10867 

17207 

47 

14361  . 

14709 

15057 

15406 

15754 

17146 

17494 

17843 

48 

14933 

15289 

15644 

16000 

16356 

17778 

18133 

18489 

49 

15517 

15880 

16243 

16000 

16968 

18420 

18783 

19146 

50 

16111 

16481 

16852 

17222 

17592 

19074 

19444 

19815 

51 

16717 

17094 

17472 

17850 

18228 

19739 

20117 

20494 

52 

173.33 

17719 

18104 

18489 

18874 

20415 

20800 

21185 

53 

17961 

1K354 

18746 

19139 

19531 

21102 

21494 

21887 

54 

18600 

19000 

19400 

19800 

20200 

21800 

22200 

22600 

55 

19250 

19657 

20005 

20472 

20880 

22509 

22917 

23324 

56 

19911 

20326 

20741 

21156 

21570 

23230 

23644 

24059 

57 

20583 

2UX)6 

21428 

21850 

23961 

24383  i 

24SU5 

^^ 

21267 

21696 

22126 

22556 

22985 

24704 

25im 

25563 

59 

21961 

22398 

22S35 

23272 

23709 

25457 

25H!>4 

2G-i3:» 

60 

22667 

23111 

23556 

24000 

24444 

26222 

260G7 

S'.'IU 

380  XIX  —CUBIC  YARDS  PER  100  FEET.     SLOPES  2  :  1. 


Depth 


1 
2 
3 
4 

5 
6 

p- 
4 

8 

9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 


31 
3-2 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
S2 
53 

54 
55 
56 
57 

58 
59 
CO 


Base 
12 


Base 
14 


Base 
16 


53 
119 
200 
296 
407 
533 
674 
830 
1000 
1185 

1385 
1600 
1830 
2074 
2.333 
2607 
2896 
3-200 
3.319 
385-2 


21 

4200 

22 

4563 

23 

4941 

24 

5333 

25 

5741 

26 

6163 

27 

6600 

28 

7052 

29 

7519 

30 

8000 

8496 

9007 

9533 

10074 

iog;3j) 

11200 
11785 
1-2385 
13)>;k) 
13630 

14274 

14' 133 
15607 
16-296 
17000 
17719 
1SJ5-2 
19-200 
19963 
20741 

81.^33 
22.^41 
23163 
24000 
24852 
25719 
26600 
27496 
28407 
39333 


59 
133 
222 
3-26 
444 
578 
726 
889 
1067 
1259 

1467 
1689 
19-26 
2178 
2444 
2726 
3022 
S;533 
3659 
4000 

4356 
4130 
5111 
5511 
5926 
6356 
6800 
7259 

I  I  CO 

82-22 

87*26 
9-244 
9:78 
10326 
1C889 
11467 
1-2059 
1-2667 
13-289 
139-26 

14578 
15244 
159-26 
16622 
1733:3 
18059 
18800 
19556 
20326 
20711 

21911 
22726 
2J556 
24400 
•25-259 
26133 
27022 
279-26 
28844 
29778 


67 
148 
244 
356 
481 
622 
778 
948 
1133 
1333 

1548 
1778 
2022 
2281 
2556 
2844 
3148 
3467 
3800 
4148 

4511 
4889 
5281 
5689 
6111 
6548 
7000 
7467 
7948 
8444 


Base 
18 


Base 
20 


14881 
155.^6 
16224 
16948 
17667 
18400 
19148 
19911 
20689 
21481 

2-2289 
23111 
23948 
24800 
25667 
26548 
27444 
28:356 
29-281 
30222 


74 

163 

267 

385 

519 

667 

&30 

1007 

1200 

1407 

16:30 
1867 
2119 
2-385 
2667 
2963 
3274 
3600 
3941 
4296 

4667 
5052 
5452 
5867 
6296 
6741 
7200 
7674 
8163 
8667 


8956 

9185 

94,S1 

9719 

10022 

10-267 

10578 

10830 

11148 

11407 

117:33 

120(X) 

1-2333 

1-2607 

12948 

13230 

l:35?8 

13867 

14222 

14519 

15185 

15867 
1656:3 
17274 

mjoo 

18741 
19496 
20267 
21052 
21852 

22667 
23496 
24:341 
252C0 
2K01'4 
26963 
27867 
28785 
29719 
30667 


81 

178 

289 

415 

556 

711 

881 

1067 

1267 

1481 

1711 
1956 
2-215 
2489 
2778 
3081 
3400 
3733 
4081 
4444 

4822 
5215 
5622 
6044 
6481 
60:33 
7400 
7881 
a378 
8SS9 

9415 
9956 
10511 
11081 
11667 
12-267 
12881 
1:3511 
14156 
14815 

15189 
16178 
16881 
17600 
ia333 
19081 
19S44 
20622 
21415 
22222 

23044 
2:3881 
24733 
25600 
26481 
27378 
28-289 
29215 
30156 
31111 


Ba-e 
28 


111 

237 

378 

53:3 

704 

889 

1089 

1304 

1533 

1778 

2037 
2311 
2600 
2904 
3222 
3556 
3904 
4267 
4644 
5037 

5444 
5S67 
C304 
6756 
7-222 
7704 
8200 
8711 
9237 
9778 

10333 
10904 
11489 
12089 
12704 
13:333 
1:3978 
146:37 
15311 
16000 

16704 
17^22 
18156 
18904 
19667 
20444 
21237 
2-2044 
22867 
23704 

24556 
25422 
26:304 
27200 
28111 
290:37 
20978 
:30933 
31904 
32889 


Base 
30 


119 

252 

400 

563 

741 

933 

1141 

i:363 

1600 

1852 

2119 
2400 
2696 
3007 
3S33 
3674 
4030 
4400 
4785 
5185 

5600 
60:30 
6474 
6933 
7407 
7896 
8400 
8919 
9452 
100(X> 

10563 
11141 
117:33 
1-2341 
1-296:3 
13600 
14252 
14919 
156<X) 
16296 

17007 
17733 
18474 
19230 
20<X)0 
20785 
21585 
22400 
2:3-2:30 
24«)74 

24933 
25807 
26690 
27600 
28519 
29452 
30400 
31363 
32341 
:33:333 


Base 
32 


126 

267 

422 

593 

778 

978 

1193 

1422 

1667 

19-26 

2200 
2489 
2793 
:3111 
3444 
3793 
4156 
4533 
4926 
5333 

5756 
6193 
6644 
7111 
7593 
8089 
8600 
9126 
9667 
10-222 

10793 
11378 
11978 
12593 
13222 
i:3867 
145-26 
15200 
15889 
16593 

17311 
18044 
18793 
19556 
203:33 
211-26 
219:3:3 
22756 
2:3593 
24444 

•25311 
•26193 
2r089 
28000 
-289-26 
29S67 
30H22 
31793 
32778 
33778 


XIX.— CUBIC  YARDS  PER  100  FEET.     SLOPES  3:  1.  381 


Depth 

Base 

Bas3 

Base 

Base 

Base 

12 

14 

16 

70 

18 

20 

1 

56 

03 

78 

85 

o 

133 

148 

163 

178 

193 

3 

233 

256 

278 

300 

322 

4 

356 

385 

415 

444 

474 

5 

500 

537 

574 

Oil 

648 

6 

667 

711 

756 

800 

844 

7 

856 

907 

959 

1011 

1063 

8 

1067 

1126 

1185 

1244 

1304 

9 

1300 

1367 

1433 

1500 

1567 

10 

1556 

1630 

1704 

1778 

1852 

11 

18.33 

1915 

1996 

2078 

2159 

12 

2133 

2222 

2311 

^00 

2489 

13 

2456 

2552 

2648 

2744 

2841 

14 

2800 

2904 

3007 

3111 

3215 

15 

3167 

3278 

3389 

3500 

3611 

16 

3556 

3G74 

3793 

3911 

4030 

17 

3967 

4093 

4219 

4344 

4470 

18 

4400 

4533 

4067 

4800 

4933 

19 

4856 

4996 

5137 

5278 

5419 

20 

5333 

5481 

5630 

5778 

5926 

21 

5833 

5989 

6144 

6300 

6456 

22 

6356 

6519 

6G81 

6844 

7007 

23 

0900 

7070 

7241 

7411 

7581 

24 

7407 

7644 

7822 

8000 

8178 

25 

8056 

8241 

8426 

8611 

8796 

26 

8667 

8859 

9052 

9244 

9437 

27 

9300 

9500 

9700 

9900 

10100 

28 

9956 

10163 

10370 

10578 

10785 

29 

10633 

10848 

11063 

11278 

11493 

30 

11333 

11556 

11778 

12000 

12222 

31 

12056 

12285 

12515 

12744 

12974 

32 

12800 

13037 

13274 

13511 

13748 

as 

13567 

13811 

14056 

14300 

14544 

M 

14356 

14607 

14859 

15111 

15363 

35 

15167 

15426 

15685 

15944 

16204 

36 

16000 

16207 

16533 

16800 

17067 

37 

16856 

17130 

17404 

17678 

17952 

38 

17733 

18015 

18296 

18578 

18859 

39 

18G33 

18922 

19211 

19500 

10789 

40 

19556 

19852 

20148 

20444 

20:^41 

41 

20500 

20S04 

21107 

21411 

21 71 5 

42 

21467 

21778 

22089 

22400 

22711 

43 

22456 

22774 

23093 

2:3411 

23730 

44 

23467 

23793 

24119 

24444 

24770 

45 

24500 

24833 

25167 

25500 

25833 

46 

25556 

25896 

26237 

26578 

26919 

47 

26633 

20981 

27330 

27678 

28026 

48 

27733  1 

28089 

28444 

28800 

29156 

49 

28S56 

29219 

2!)581 

29944 

30307 

50 

30000 

30370 

30741 

31111 

31481 

51 

31167 

31544 

31922 

32300 

32678 

52 

32356 

32741 

33126 

33511 

33896 

53 

33567 

a3959 

34352 

34744 

a5137 

54 

34800 

35200 

35000 

36000 

36400 

55 

36056 

36463 

36870 

37278 

37685 

56 

371333 

37748 

38163 

38578 

38993 

57 

386*3 

39056 

39478 

39900 

40322 

58 

39956 

40385 

40815 

41244 

41074 

59 

41300 

41737 

42174 

42611 

43048 

60 

42667 

43111 

43556 

44000 

44444 

Base 
28 

115 

252 

411 

693 

796 

1022 

1270 

1541 

1833 

2148 

24a5 
2844 
3226 
3630 
4056 
4504 
4974 
5467 
5981 
6519 

7078 

7659 

8263 

8889 

9537 

10207 

10900 

11615 

12352 

13111 

13893 
14696 
15522 
16370 
17241 
18ia3 
19048 
19985 
20944 
21926 

22930 
23956 
25004 
26074 
27167 
28281 
29419 
30578 
31759 
32963 

34189 
35437 
36707 
38000 
39315 
40652 
42011 
43393 
4^798 
46222 


Base 

Base 

30 

32 

122 

130 

267 

281 

433 

456 

622 

652 

833 

870 

1067 

nil 

1322 

1374 

1600 

1659 

1900 

1967 

2222 

2296 

2567 

2648 

2933 

3022 

3322 

3419 

3733 

3837 

4167 

4278 

4622 

4741 

5100 

5226 

5600 

5733 

6122 

6263 

6667 

6815 

7233 

7389 

7822 

7985 

8433 

8504 

9067 

9144 

9722 

9807 

10400 

10593 

11100 

11300 

11822 

12030 

12567 

12781 

13333 

13556 

14122 

14352 

149*3 

15170 

15767 

16011 

16622 

16874 

17500 

17759 

18400 

18667 

19322 

19596 

20267 

20548 

21233 

21522 

^fV<^i^/V 

22516 

23233 

23537 

24267 

24578 

25322 

25641 

26400 

26726 

27500 

27833 

28622 

28963 

29767 

30115 

30933 

31289 

32122 

32485 

333*3 

33704 

a4567 

34944 

35822 

36207 

37100 

37493 

38400 

38800 

39722 

40130 

41067 

41481 

42433 

42856 

43822 

44252 

45233 

45670 

46667 

47111 

382   TABLE  XX. -CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area 

Cubic 

Area. 

Cubic 

Area. 
Sq. 
Ft. 

Cubic  1 

Area. 

Cubic 

Area . 
Sq. 
Ft. 

Cubic 

Sq. 
Ft. 

1 

Yards. 

Sq. 
Ft. 

Yards. 

Y'ards. 

Sq. 
Ft. 

Yards. 

Yards. 

3.7 

51 

188.9 

101 

{ 

.374.1 

151 

5.59  3 

201 

744  4 

2 

7.4 

52 

192.6 

102 

377.8 

1.52 

.563.0 

202 

748.2 

3 

11.1 

53 

196.3 

103 

381.5 

153 

566.7 

203 

751.9 

4 

14.8 

54 

200.0 

104 

;i85.2 

151 

.570.4 

204 

7.55.6 

5 

18.5 

55 

203.7 

105 

388  9 

155 

.574.1 

205 

759.3 

6 

22.2 

56 

207.4 

106 

392.6 

1.56 

577.8 

206 

763.0 

1 

25.9 

57 

211.1 

107 

396.3 

157 

581.5 

207 

766.7 

8 

29.6 

58 

214.8 

108 

400  0 

1.58 

.585.2 

208 

770.4 

9 

33.3 

59 

218.5 

109 

403.7 

159 

588.9 

209 

774.1 

10 

37.0 

60 

222.2 

110 

407.4 

160 

.592.6 

210 

777.8 

11 

40.7 

61 

225.9 

111 

411.1 

161 

596.3 

!     211 

781  5 

1-,' 

4-1.4 

62 

229.6 

112 

414.8 

162 

600.0 

212 

785.2 

13 

48.1 

63 

233.3 

113 

418.5 

163 

603.7 

!     213 

788.9 

14 

51.9 

64 

237.0 

114 

422  2 

164 

607.4 

'     214 

792.6 

15 

55.6 

65 

240.7 

115 

425^9 

165 

611. 1 

215 

796.3 

16 

59.3 

66 

244.4 

116 

429.6 

166 

614  8 

216 

800.0 

IT 

63.0 

67 

248.2 

117 

433.3 

167 

618.5 

i     217 

803.7 

18 

66.7 

68 

251.9 

118 

4.37.0 

168 

622.2 

218 

807.4 

19 

70.4 

69 

2.55.6 

119 

440.7 

169 

625.9 

'     219 

811.1 

20 

74.1 

70 

2.59.3 

I     120 

444.4 

170 

629.6 

1     220 

814.8 

21 

77.8 

71 

263.0 

■     121 

448.2 

171 

633.3 

221 

818.5 

22 

81.5 

72 

266.7 

i     122 

451.9 

172 

637.0 

1     222 

822.2 

23 

85.2 

73 

270.4 

123 

455  6 

173 

640.7 

223 

825.9 

24 

88.9 

74 

274.1 

124 

4.59.3 

174 

644.4 

224 

829.6 

2.5 

92  6 

75 

277.8 

125 

4r,3.0 

175 

648.2 

225 

833.3 

28 

96.3 

76 

281.5 

126 

466.7 

176 

651.9 

226 

837.0 

27 

100.0 

77 

285  2 

127 

470.4 

177 

6.55.6 

227 

840.7 

28 

103.7 

78 

288.9 

128 

474  1 

178 

6.59.3 

228 

844.4 

29 

107.4 

79 

292.6 

129 

477.8 

179 

663.0 

229 

84S.2 

:30 

111.1 

80 

296.3 

130 

481.5 

180 

666.7 

230 

8.51.9 

31 

114.8 

81 

300.0 

131 

485.2 

181 

670.4 

231 

8.55.6 

32 

118.5 

82 

303.7 

1.32 

488  9 

182 

674.1 

232 

859.3 

33 

122.2 

83 

307.4 

133 

492.6 

183 

677.8 

2:33 

863.0 

34 

125.9 

84 

311.1 

134 

496.3 

184 

681.5 

234 

866.7 

35 

129  6 

85 

314.8 

i:« 

500.0 

185 

685  2 

235 

870.4 

36 

133.3 

86 

318.5 

136 

503  7 

186 

688.9 

236 

874.1 

37 

137.0 

87 

322.2 

137 

.507.4 

187 

692.6 

237 

877.8 

38 

140.7 

88 

325.9 

138 

511.1 

188 

696  3 

238 

881.5 

39 

144  4 

89 

329.6 

139 

514.8 

189 

700.0 

2S9 

885.2 

40 

.148.2 

90 

333.3 

140 

518.5 

190 

703  7 

240 

888.9 

41 

151.9 

91 

387. 0 

141 

522.2 

191 

707.4 

241 

892.6 

42 

155.6 

92 

340.7 

I     142 

.525.9 

192 

711.1 

242 

896.3 

43 

159  3 

93 

.344  4 

143 

529.6 

193 

714.8 

243 

900.0 

44 

163.0 

94 

348.2 

144 

.5.33  3 

194 

718.5 

244 

903.7 

45 

166.7 

95 

3.51.9 

i     145 

.537.0 

195 

722.2 

245 

907.4 

46 

170.4 

96 

3.55.6 

'     146 

540.7 

196 

725.9 

246 

911.1 

47 

174.1 

97 

3.59.3 

1     147 

.544.4 

197 

729.6 

'     247 

914.8 

48 

177.8 

G8 

363.0 

1     148 

.548.2 

198 

733.3 

248 

918.5 

49 

181.5 

99 

366.7 

!     149 

.5.51.9 

199 

737.0 

249 

9C2.2 

50 

185.2 

100 

370.4 

150 

555.6 

200 

740.7 

250 

925.9 

TABLF>  XX.— CUBIC  YARDS  IN  100  FEET  LENGTH.  383 


Area. 

Cubic 

Area. 
Sq. 
Ft. 

Cubic 

Area. 
Sq. 
Ft. 

Cubic 

Area. 

&^- 
Ft. 

Cubic 

Area. 
Sq. 
Ft. 

Cubic 

Sq. 
Ft. 

Yards. 

Yards. 

Yards. 

Yards. 

Yards. 

2.51 

929.6 

301 

1114.8 

351 

1300.0 

401 

1485.2 

451 

1670.4 

253 

933.3 

302 

1118.5 

352 

1303.7 

402 

1488.9 

152 

1674.1 

2.53 

937.0 

303 

1122.2 

353 

1307.4 

403 

1492.6 

453 

1677.8 

2,51 

940.7 

304 

1125.9 

354 

1311.1 

404 

1496.3 

4.54 

1681.5 

25.5 

944.4 

305 

1129.6 

355 

1314.8 

405 

1500.0 

4.55 

1685.2 

256 

948  2 

306 

1133.3 

3.56 

1318.5 

406 

1.503.7 

4.56 

1688.9 

257 

951.9 

307 

1137.0 

357 

1322.2 

407 

1507.4 

457 

1692.6 

258 

955.6 

308 

1140.7 

358 

1325.9 

408 

1511.1 

458 

1696.3 

250 

9.59.3 

309 

1144.4 

359 

1329.6 

409 

1514.8 

459 

1700.0 

260 

963.0 

310 

1148.2 

360 

1333.3 

410 

1518.5 

460 

1703.7 

261 

966.7 

311 

1151.9 

361 

1337.0 

411 

1522.2 

461 

1707.4 

262 

970.4 

312 

1155.6 

362 

1340.7 

412 

1525.9 

462 

1711.1 

263 

974.1 

313 

11.59.3 

363 

1344.4 

413 

1529.6 

463 

1714.8 

264 

977.8 

314 

1163.0 

364 

1348.2 

414 

1533.3 

464 

1718.5 

265 

981.5 

315 

1166.7 

365 

1351.9 

415 

1537.0 

465 

1722.2 

266 

985.2 

316 

1170.4 

366 

1355.6 

416 

1540.7 

466 

1725.9 

26? 

988.9 

317 

1174.1 

367 

1359.3 

417 

1544.4 

467 

1729.6 

268 

992.6 

318 

1177.8 

368 

1363.0 

418 

1.548.2 

468 

1733.3 

26!) 

996.3 

319 

1181.5 

369 

1366.7 

419 

1551.9 

469 

1737.0 

270 

1000  0 

320 

1185.2 

370 

i:i70.4 

420 

1555.6 

470 

1740.7 

271 

1U03.7 

321 

1188.9 

.371 

1374.1 

421 

15.59.3 

471 

1744.4 

272 

1007.4 

322 

1192.6 

372 

1377.8 

422 

1563.0 

472 

1748.2 

273 

1011.1 

323 

1196.3 

373 

1381.5 

423 

1566.7 

473 

1751.9 

271 

1014  8 

324 

1200  0 

374 

1385.2 

424 

1570.4 

474 

1755.6 

275 

1018.5 

325 

1203.7 

375 

1388.9 

425 

1574.1 

475 

1759.3 

276 

1022.2 

326 

1207.4 

376 

1392.6 

426 

1577.8 

476 

1763.0 

277 

1025.9 

327 

1211.1 

377 

1396.3 

427 

1581.5 

477 

1766.7 

278 

1029.6 

328 

1214.8 

378 

1400.0 

428 

1585.2 

478 

1770.4 

279 

1033.3 

.329 

1218.5 

379 

1403. 7 

429 

1588.9 

479 

1774.1 

280 

10,37.0 

330 

1222  2 

380 

1407.4 

430 

1592.6 

480 

1777.8 

281 

1040.7 

331 

1225.9 

381 

1411.1 

431 

1596.3 

481 

1781.5 

282 

1044.4 

332 

1229  6 

382 

1414.8 

432 

1600.0 

482 

1785.2 

283 

1048.2 

333 

1233.3 

383 

1418.5 

433 

1603.7 

483 

1788.9 

284 

10.51.9 

334 

1237.0 

384 

1422.2 

434 

1607.4 

484 

1792.6 

285 

1055.6 

335 

1240.7 

385 

1425.9  ; 

435 

1611.1 

485 

1796.3 

286 

10.59.3 

336 

1244.4 

386 

1429.6  1 

436 

1614.8 

486 

1800.0 

287 

1063.0 

337 

1248.2 

387 

1433.3  1 

437 

1618.5 

487 

1803.7 

288 

1066.7 

338 

1251.9 

388 

1437.0  1 

438 

1622.2 

488 

1807.4 

289 

1070.4 

339 

12.55.6 

389 

1440.7 

439 

1625.9 

i89 

1811.1 

290 

1074.1 

340 

12.59.3 

390 

1444.4 

440 

1629.6 

490 

1814.8 

291 

1077.8 

341 

1263.0 

391 

1448.2 

441 

1633.3 

491 

1818.5 

292 

1081.5 

342 

1266.7 

392 

1451.9 

442 

1637.0 

492 

1822.2 

293 

1085.2  : 

343 

1270.4 

393 

14.55.6 

443 

1640.7 

493 

1825.9 

294 

1088.9 

344 

1274.1 

394 

14.59.3 

444  1 

1644.4 

494 

1829.6 

295 

1092.6 

.345 

1277.8 

395 

1463.0 

445 

1648  2 

495 

1833.3 

296 

1096.3 

346 

1281.5 

396 

1466.7 

446 

1651.9 

496 

18.37.0 

297 

1100.0 

347 

1285.2 

397 

1470.4 

447  1 

1655.6 

497 

1840.7 

298 

1103.7 

348 

1288.9 

398 

1474.1 

448 

16.59.3 

498 

1844.4 

299 

1107.4 

349 

1292.6 

399 

1477.8 

449 

1663.0 

499 

1818.2 

300 

1111.1 

350 

1296.3 

400 

1 

1481.5 

450 

1666.7 

500 

1851.9 

rXTW-AW    •     ■!   »-  •■n 


384  TABLE  XX.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 
Sq. 
Ft. 

501 

Cubic 

Area. 
Sq. 
Ft. 

Cubic  1 

Area. 
Sq. 
Ft. 

Cubic 

Area. 
Ft. 

1 

Cubic 

Area. 
Ft. 

Cubic 

Yards. 

Yards. 

Yai-ds. 

Yards. 

Yard?. 

1855.6 

551 

2040.7 

601 

2225.9 

651 

1 

2411.1 

701 

2596.3 

502 

1859.3 

552 

2044.4 

602 

2229.6 

652 

2414.8 

702 

2600.0 

503 

1863.0 

5.53 

2048.2 

603 

2233.3 

653 

2418.5 

703 

2603.7 

504 

1866.7 

5.54 

2051.9 

604 

2237.0 

654 

2422.2 

704 

2607.4 

505 

1870.4 

555 

2055.6 

605 

2240.7 

655 

2425.9 

705 

2611.1 

506 

1874.1 

556 

20.59.3 

606 

2244.4 

656 

2429.6 

706 

2614.8 

507 

1877.8 

557 

2063.0  i 

607 

2248.2 

657 

2433.3 

707 

2618.5 

508 

1881.5 

558 

2066.7 

608 

2251.9 

658 

2437.0 

708 

2622.2 

.509 

1885.2 

559 

2070.4 

609 

2255.6 

659 

2440.7   1 

709 

2625.9 

510 

188S.9 

560 

2074.1 

610 

2259.3 

660 

2444.4 

710 

2629.6 

.511 

1892.6 

.561 

2077.8 

611 

9263  0 

661 

2448.2 

711 

2633.3 

512 

1896.3 

562 

2081.5 

612 

2266.7 

662 

2451.9 

712 

2637.0 

5i:^ 

1900.0 

563 

2085.2 

613 

2270.4 

663 

2455.6 

713 

2640.7 

514 

1903.7 

564 

2088.9 

614 

2274.1 

664 

2459.3 

714 

2644.4 

.515 

1907.4 

5(35 

2092.6 

615 

2277.8 

665 

2463.0   i 

715 

2648.2 

516 

1911.1 

566 

2096.3 

616 

2281.5 

666 

2466.7   1 

716 

2651.9 

51 : 

1914.8 

567 

2100.0 

617 

2285.2 

667 

2170.4 

717 

2655.6 

518 

1918.5 

568 

2103.7 

618 

2288.9 

668 

•2474.1    1 

718 

2659.3 

519 

19-22.2 

569 

2107.4 

619 

^292.6 

669 

2477.8   ! 

719 

2663.0 

.520 

1925.9 

570 

2111.1 

620 

2296.3 

670 

2481.5   i 

720 

2666.7 

521 

1929  6 

571 

2114.8 

621 

2300.0 

671 

2485.2 

721 

2670.4 

522 

1933.3 

572 

2118.5  1 

622 

2.303.7 

672 

2488.9 

722 

2674.1 

523 

19.37.0 

573 

2122.2  ' 

623 

2307.4 

673 

2492  6  ; 

723 

2677.8 

524 

1940.7 

574 

2)25.9 

624 

2311.1 

674 

2496.3 

724 

2681.5 

525 

1944.4 

575 

2129.6 

625 

2314.8 

675 

■2500.0 

725 

2685.2 

526 

1948.2 

576 

2133.3 

626 

2318.5 

676 

2503.7 

726 

2688.9 

.527 

1951.9 

577 

2137.0 

627 

2322.2 

677 

2507.4 

727 

2692.6 

528 

1955.6 

578 

2140.7 

628 

2325.9 

678 

2511.1 

728 

2696.3 

529 

1959.3 

579 

2144.4 

629 

2329.6 

679 

2514.8 

729 

2700.0 

530 

1963.0 

580 

2148  2  : 

630 

2333.3 

680 

2518.5 

730 

2703.7 

531 

1966.7 

581 

2151.9  j 

631 

2337.0 

681 

2522.2 

731 

2707.4 

532 

1970.4 

582 

21.55.6 

632 

2*40.7 

682 

2525.9 

732 

2711.1 

533 

1974.1 

583 

21.59.3 

633 

2344.4 

683 

2529.6 

733 

2714.8 

534 

1977.8 

584 

2163.0  i 

634 

2348.2 

684 

2533.3 

734 

2718.5 

535 

1981.5 

585 

2166.7  , 

635 

2:3.51.9 

685 

2537.0 

735 

2722.2 

536 

1985.2 

586 

2170.4 

636 

23.55.6 

686 

2540  7 

736 

2725.9 

537 

1988.9 

587 

2174.1 

637 

23.59.3 

687 

2544.4 

737 

2729.6 

538 

1992.6 

588 

2177.8 

638 

2363  0 

688 

2.548.2 

738 

2733.3 

539 

1996.3 

589 

2181.5 

639 

2366.7 

689 

2.551.9 

739 

2737.0 

540 

2000.0 

590 

2185.2 

640 

2370.4 

690 

2.555.6 

740 

2740.7 

.541 

2003.7 

591 

2188.9 

641 

2374.1 

691 

2.5.59.3 

741 

2744.4 

542 

2007.4  ■: 

592 

2192.(! 

642 

2377.8 

692 

2563.0 

742 

2748.2 

543 

2011.1 

593 

2196.3 

643 

2381.5 

693 

2566.7 

743 

2751.9 

544 

2014.8 

594 

2200.0 

644 

2.385.2 

694 

2.570.4 

744 

2755.6 

545 

2018.5 

595 

2203.7 

645 

2388.9 

695 

2.574.1 

745 

27.59.3 

546 

2022.2 

596 

2207.4 

646 

2.392.6 

696 

2577.8 

746 

2763.0 

547 

2025.9 

597 

2211.1 

647 

2396.3 

697 

2581.5 

747 

2766.7 

548 

2029.6 

598 

2214.8 

648 

2400.0 

698 

2.585.2 

748 

2770.4 

549 

2033.3 

599 

2218.5 

649 

2403.7 

699 

2588.9 

749 

2774.1 

550 

2037.0 

600 

2222.2 

650 

2407.4 

700 

2592.6 

750 

2777.8 

TABLE  XX.— CUBU;  YARDS  IN  100  FEET  LENGTH.  385 


Area . 

Cubic 

Area. 

Sq. 
Ft. 

Cubic 

Area. 

Sq. 
1    Ft. 

Cubic 

Area. 
Sq. 
Fi. 

Cubic 

Area. 
Sq. 
Ft. 

Cubic 

Sq. 
Ft. 

Yards. 

Yards. 

Yards. 

I 

Yards. 

Yards. 

751 

2781.5 

801 

2966.7 

!     851 

3151.9 

901 

3337.0 

951 

3522.2 

75-^ 

:^785.^> 

802 

2970.4 

852 

3155.6 

902 

3340.7 

952 

3525.9 

753 

2788.9 

80? 

2974.1 

!     853 

3159.3 

903 

3344.4 

953 

3529.6 

754 

2792.6 

8K4 

2977.8 

854 

3163.0 

904 

3348.2 

954 

3533.3 

755 

2796.3 

805 

2981.5 

855 

3166.7 

905 

3351 . 9 

955 

3537.0 

75'3 

2800.0 

806 

2985.2 

1     856 

3170.4 

906 

3355.6 

956 

3540.7 

757 

2803.7 

807 

2988.9 

857 

3174.1 

907 

3359.3 

957 

3544.4 

758 

2807.4 

808 

2992.6 

858 

3177.8 

908 

3363.0 

958 

3548.2 

759 

2811.1 

809 

2996.3 

!     859 

3181.5 

909 

3366.7 

959 

3551  9 

760 

2811.8 

810 

3000.0 

i     860 

3185.2 

910 

3370.4 

960 

3555.6 

761 

2818.5 

811 

3003  7 

861 

3188.9   ; 

911 

3374.1 

961 

35.59.3 

76-.> 

2822  2 

812 

3007.4 

862 

3192.6 

912 

3377.8 

962 

3563.0 

763 

2825.9 

813 

3011.1 

863 

3196.3 

913 

3381.5 

963 

3566.7 

764 

2829.6 

814 

3014.8 

864 

3200.0 

914 

3385.2 

964 

.%70.4 

705 

2833  3 

815 

3018.5 

865 

3203.7 

915 

3388.9 

965 

3574.1 

766 

2837.0 

816 

3022.2 

866 

3207.4 

916 

3392.6 

966 

3577.8 

767 

J840.7 

817 

3025.9 

\     867 

321 1 . 1 

917 

3396.3 

967 

3581.5 

768 

2844.4 

818 

3029.6 

868 

3214.8 

918 

3400.0   i 

968 

3585.2 

769 

2848.2 

819 

3033.3 

i     869 

3218.5   i 

919 

3403.7 

969 

3588.9 

770 

2851.9 

820 

30S7.0 

!     870 

3222.2   ' 

920 

3407.4 

970 

3592.6 

771 

J855.6 

821 

3040.7 

1     871 

3225.9 

921 

3411.1 

971 

3596.3 

77i 

2859.3 

822 

3044.4 

!     872 

3229.6 

922 

3414.8 

972 

3600.0 

773 

2863.0 

823 

3048.2 

873 

3233.3 

923 

3418.5 

973 

3603.7 

771 

2866.7 

824 

3051.9 

-     874 

3237.0   ' 

924 

3422.2 

974 

3607.4 

775 

2870.4 

825 

3055.6 

1     875 

3240.7 

925 

3425.9 

975 

3611.1 

776 

2874.1 

826 

3059.3 

876 

3244.4 

926 

3429.6 

976 

3614.8 

1  1  1 

2877.8 

827 

3063.0 

877 

3248.2 

927 

3433.3 

977 

3618.5 

77K 

2881.5 

828 

3066.7 

j     87'8 

3251.9 

928 

3437.0 

978 

3622.2 

779 

2885.2 

829 

3070.4 

!     879 

3255.6 

929 

3440.7 

979 

3625.9 

780 

2888.9 

830 

3074.1 

;     880 

3259.3 

930 

3444.4 

980 

3629.6 

781 

2892.6 

831 

3077.8 

881 

3263.0   ' 

931 

3448.2 

981 

3633.3 

78:i 

2896.3 

832 

3081.5 

882 

3266.7 

932 

3451.9 

982 

3637.0 

783 

2900.0 

833 

3085.2 

1     883 

3270.4 

933 

3455.6 

983 

3640,7 

784 

290^  7 

834 

3088.9 

884 

3274.1 

934 

3459.3 

984 

3644.4 

785 

2907.4 

835 

3092.6 

1     885 

3277.8 

935 

3463.0 

985 

3648.2 

786 

4911.1 

836 

3096  3 

:    886 

3281.5 

936 

3466.7 

986 

3651.9 

787 

2914.8 

837 

3100.0 

:    887 

3285.2 

937 

3470.4 

987 

3655.6 

788 

2918.5 

838 

3103.7 

1    888 

3288.9 

938 

3474.1 

988 

3659.3 

789 

2922.2 

839 

3107.4 

!    889 

3292.6 

939 

3477.8 

989 

3663.0 

790 

2925.9 

840 

3111.1 

!     890 

3296.3 

940 

3481.5 

into 

3666.7 

791 

2929.6 

841 

3114.8 

'     891 

3300.0 

941 

3485.2 

991 

3670.4 

79-3 

2933.3 

842 

3118.5 

892 

3303.7 

942 

3488.9 

992 

3674.1 

793 

2937.0 

843 

3122.2 

893 

3307.4 

943 

3492.6 

993 

3677.8 

794 

2940.7 

844 

3125.9 

894 

3311.1 

944 

3496.3 

994 

3681.5 

795 

2944.4 

845 

3129.6 

895 

3314.8 

945 

3.'-.00.0 

995 

3685.2 

796 

2948.2 

846 

3133.3 

896 

3318.5 

946 

3.503.7 

996 

3688 . 9 

797 

2951.9 

847 

3137.0 

897 

3322.2 

947 

3507.4 

997 

3692.6 

798 

29.55.6 

848 

3140.7 

898 

3325,9 

948 

3511.1 

998 

3696.3 

799 

29.59.3 

849 

3144.4 

i     899 

3329.6 

949 

3.514.8 

999 

3700.0 

800 

2963.0 

850 

3148.2 

900 

3333.3   ; 

950 

3518.5 

1000 

3703.7 

rz^^^AM  * 


386       XXI.— RISE  PER  MILE  OF  VARIOUS  GRADES. 


Rise 

per 

Cent. 


.01 
.02 
.03 
.04 
.05 
.06 
.07 
.08 
.09 
.10 

.11 
.12 
.13 
.14 
.15 
.16 
.17 
.18 
.19 
.20 

.21 
.22 
.23 
.24 
.25 
.26 
.27 
.28 
.29 
.30 

.31 
.32 
.33 
.34 
.35 
.36 
.37 
.38 
.39 
.40 

.41 
.42 
.43 
.44 
.45 
.46 
.47 
.48 
.49 
.50 

.51 
.52 
.53 
.54 
.55 
.56 
.57 
.58 
.59 
.60 


Rise 

Feet  per 
3Iile. 

per 
Cent. 

.528 

.61 

1.056 

.62 

1.5S4 

.63 

2.112 

.64 

2.640 

.65 

3.168 

.66 

3.696 

.67 

4.224 

.68 

4.752 

.69 

5.280 

.70 

5.808 

.71 

6.336 

.72 

6.864 

.73 

7.392 

!       .74 

7.920 

.75 

8.448 

.76 

8.976 

.77 

9.504 

.78 

10.032 

.79 

10,560 

.80 

11.088 

.81 

11.616 

.82 

12.144 

.83 

12.672 

.&4 

13.200 

.85 

13.728 

.86 

14.256 

.87 

14.784 

.88 

15.312 

.89 

15.840 

.CO 

16.368 

.91 

16.896 

.92 

17  424 

.93 

17.952 

.94 

18.480 

.95 

19.008 

.96 

19.5.36 

.97 

20.064 

.98 

20.592 

.99 

21.120 

1.00 

21.648 

1.01 

22.176 

1.02 

22.704 

•    1.03 

23.232 

1.04 

23.760 

1.05 

24.288 

1.06 

24.816 

1     1.07 

25.344 

1.08 

25.872 

I     1.09 

26.400 

1.10 

26  928 

1.11 

27.456 

1.12 

27.984 

1.13 

28.512 

1.14 

29.040 

1.15 

29.568 

1.16 

30.096 

1.17 

30.624     ' 

1.18 

31   152 

1.19 

31.080 

i     1.20 

1 
Feet  per 
Mile.      1 

1 

Rise 

per 

Cent. 

32.208 

1.21 

32.736 

1     1.22 

33  264 

1     1.23 

33.792 

1.24 

34.320 

1.25 

.34.848 

1.26 

35.. 376 

1.27 

35.904 

1     1.28 

36.432 

1.29 

36.900     1 

1 

1.30 

37.488 

1.31 

38.016 

1.32 

38.544 

1.33 

39.072 

1..34 

39.600 

1.35 

40.128 

1.36 

40.656 

1.37 

41.184 

1.38 

41.712 

1.39 

42.240 

1.40 

42.768 

1.41 

43.296     1 

1.42 

43.824     ! 

1.43 

44.3.52 

1.44 

44.880 

;     1  45 

45.408 

1     1.46 

45.936 

1.47 

46.464 

1.48 

46.992 

1.49 

47.520 

1.50 

48.048 

1.51 

48.576 

1..52 

49.104 

1     1..53 

49.632 

1.54 

50.160 

1.55 

50.688 

1.56 

51.216 

1.57 

51  744 

1.58 

52.272 

1.59 

52.800 

1.60 

53.328 

1.61 

53.8.56 

1.62 

5  4.. 384 

1.63 

54.912 

1.64 

55.440 

1.65 

55.908 

1.66 

.56.496 

1.67 

57.024 

1  68 

57.552 

1.69 

58.080 

1.70 

58.608 

1.71 

.59.i:36 

1.72 

.59.664 

1.73 

60.192 

1.74 

60  720 

;     1.75 

61.248 

1     1.76 

61.776 

1.77 

62  304 

1  78 

62.^32 

1.79 

63.360     1 

l.SO 

Feet  per  j 

'     Rise 
1      -^^^ 

Mile. 

'     per 
<    Cent. 

63.888 

1     1.81 

64  416 

1.82 

64.944 

1.83 

65.472 

1.84 

66.000 

1.85 

66.528 

1.86 

67.056 

1.87 

67.584 

\     1.88     i 

68.112 

1.89 

68.640 

1.90 

69.168 

1  91 

69.696 

1.92 

70.224 

1.93 

70.752 

1.94 

71.280 

1.95 

71.808 

1.96 

72.336 

1.97 

72.864 

1.98 

73.392 

1.99     i 

73.920 

2  00     ; 

74.448 

2.10 

74  976 

2.20 

75.504 

2.30     ! 

76.032 

2.40 

76.560 

2.50 

77.088 

2.ti0     i 

77.616 

2.70 

78.144 

2  80 

78.672 

2.90 

79.200 

3.00 

79.728 

3.10 

80.2.56 

3.20 

80.784 

3.30     ' 

81.312 

3.40 

81.840 

3.50 

82.368 

3.60     . 

82.896 

3.70 

83.424 

3  80 

83.952 

3.90 

84.480 

4.00 

85.008 

4.10 

85.5;B6 

4.20 

86.064 

4.30 

86.592 

4.40 

87.120 

4.50 

87.648 

4.60 

88.176 

•     4.70 

88.704 

4.80 

89.232 

4.90 

89.760 

5.00 

90.288 

5.10 

90  816 

5.20 

91.344 

5.30 

91.872 

5.40 

92.400 

5  50 

92.928 

5.60 

93.4.56 

5.70 

93.984 

5  80 

94.512 

5.90 

95.040     , 

6.00 

Feet  per 
Mile. 


95.568 
96.096 
96.624 
97.152 
97.680 
98.208 
98.736 
99.264 
99.792 
100.320 

100.848 
101.376 
101.904 
102.432 
102.960 
103.488 
104.016 
104.544 
105.072 
105.600 

110.880 
116.160 
121.440 
126.720 
132.000 
137.280 
142.560 
147.840 
153,120 
158.400 

163.680 
168.960 
174.240 
179  520 
184.800 
190.080 
195.360 
200,640 
205.920 
211.200 

216.480 
221.760 
227.040 
232,. 320 
237.600 
242.880 
248.160 
252.440 
258.720 
264.000 

269.280 
274.560 
279.840 
285.120 
290.400 
295.680 
.300.960 
.306.240 
311.520 
316.800 


TABLE   XXII.— SLOPES   FOR   TOPOGRAPHY.       387 


fl 

<i>     ^ 

od 

a 

«     -; 

od 

a 

a>     -i 

od 

o 

CO       cS 

*-•  y-l 

o 

2     .5 

o 

W          OS 

boa 

_      o 
•^-^  o 

0)  •- W 

r2  ©«-i 

o5  c3 

-imt 

o  c 
tea 

1    a     a 

•^'-  o 

-  Cffi 

c  2  n 

O  at  V3 

•,-.'^(5 
oQ  aJ 

u-t  oj 
O  C 

(5     a 
— ,      o 

'SO 

«y  o 

a  B  ^ 
o5.2 
—  ^05 
oQoj 

< 
0=  20' 

> 

w 

<J 

> 

K 

!  < 

> 

ffi 

.58 

1 

1718.9 

1 

7°  30' 

12.87 

i  t  ,  t 

16° 

28.67 

34.9 

40 

1.16 

859.4 

40 

13.46 

74.3 

17 

30.57 

32.7 

] 

1.75 

572.9 

8 

14.05 

71.2 

18 

32.49 

30.8 

20 

2.33 

429.6 

20 

14.65 

68.3 

19 

34.43 

29.0 

40 

2.91 

343.7 

40 

15.24 

65.6 

20 

36.40 

27.5 

2 

3.49 

286.4 

9 

15.84 

63.1 

21 

38.39 

26.1 

20 

4.08 

245.4 

20 

16.44 

60.8 

22 

40.40 

24.8 

40 

4.66 

214.7 

40 

17.03 

58.7 

23 

42.45 

23.6 

3 

5.24 

190.8 

10 

17.63 

56.7 

24 

44.52 

22.5 

20 

5.83 

171.7 

20 

18.23 

54.8 

25 

46.63 

21.4 

10 

6.41 

156.0 

40 

18.84 

53.1 

26 

48.77 

20.5 

4 

6.99 

143.0 

11 

19.44 

51.4 

27 

50.95 

19.6 

20 

7.. 58 

133.0 

30 

20.35 

49.2 

28 

53.17 

18.8 

40 

8.16 

122.5 

12 

21.26 

47.0 

29 

55.43 

18.0 

5 

8.75 

114.3 

30 

23.17 

45.1 

30 

57.74 

17.3 

20 

9.34 

107.1 

13 

23.09 

43.3 

m 

70.02 

14.3 

40 

9.92 

100.8 

30 

24.01 

41.7 

40 

83.91 

11.9 

6 

10.51 

95.1 

14 

24.93 

40.1 

45 

100.00 

10.0 

20 

11.10 

90.1 

30 

25.86 

38.7 

50 

119.18 

8.4 

40 

11.69 

85.6 

15 

36.79 

37.3 

55 

142.81 

7.0 

7 

12.28 

81.4     i 

1 

30 

27.73 

36.1 

60 

173  21 

5.8 

TABLE 

XXIII.- 

-MATERIAL   ] 

REQUIRED  FOR  ONE   MILE  OF  TRACK. 

RAIL   WEIGHTS. 

RAILROAD   SPIKES.                            ] 

o 

4-1 

o 

«J      . 
C   CO 
OjO 

•Ck> 

Required  for 

OJ 

u 

3  S5 

^  be 
I)  a^  73 

5*  • 

Ties  3  ft.  Apart. 

For  Rails 
Weighing 

'sS) 

a  OS 

t-O 

bC-v 

...  '^ 

-  7;o 

O  jj 

5>^ 

.§- 

a  oi 

UK 

9J  OJo 

oW 

Pi 

02 

1-3 

02 

< 

!^.S 

^ 

12 

21.12 

18.857 

^iXl% 

360 

5870 

29.3 

45  to  70  lbs. 

16 

28.16 

35.148 

5  Xt«b 

400 

5380 

26.4 

40  "  56     ' 

20 

35.30 

31  429 

5  X   i 

450 

4690 

23.5 

35  "  40    " 

25 

44.00 

39  386 

4iX  i 

530 

3980 

19.9 

28  "  35    ' 

30 

.53.80 

47.143 

4  X  i 

600 

3530 

17.6 

24  "  35     ' 

35 

61.60 

55.000 

4iXT'B 

680 

3110 

15.5 

20  "  30     ' 

40 

70.40 

63.857 

4  X/b 

720 

3930 

14.7 

20  "  30    ' 

45 

79.20 

70.714 

3^X/5 

900 

3350 

11.7 

16  "  35     ' 

56 

98.. 56 

88.000 

4X1 

1000 

3110 

10.6 

16  "  35     ' 

60 

105.60 

94.386     > 

34  X  f 

1190 

1770 

8.9 

16  "  20     ' 

70 

133.20 

110.000 

3X1 

1240 

1700 

8.5 

16  "  20     ' 

80 

140.80 

125.714 

2iX  f 

1342    !     1570 

7.9 

12  "  16     "    , 

'number  of  splice-joints.    ' 

NUMBER 

OF  CROSS-TIES. 

Two  Bars  with  Four  Bolts  and 
Nuts  to  Each  Joint.             ! 

Distanoe  apa 

irt,  c.  to  c,  in  Feet. 

Length  of  Rail  in  Feet. 

1.5 

1.75 

2.0 

2.25 

2.50 

20 

24 

26 

28 

30 

3520 

3017 

3640 

2347 

3112 

528 

440 

406 

377 

352 

388 


TABLE   XXIV. 


CONVERSION 

OF  ENGLISH 

INCHES  INTO  CENTIMETRES 

• 

Ins. 

0 

1 

2 

3 

4 

5 

6 

i 

8 

9 

0 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 

Cm. 

0.000 

25.40 

50.80 

76.20 

101.60 

127.00 

152.40 

177.80 

203.20 

228.60 

254.00 

Cm. 

2.540 

27.94 

53.34 

78.74 

104.14 

129.54 

154.94 

180.34 

205.74 

231.14 

256.54 

Cm. 

5.080 

30.48 

55.88 
81.28 
106.68 
132.08 
157.48 
182.88 
208.28 
233.68 
2.59.08 

Cm. 

7.620 

33.02 

58.42 

83.82 

109.22 

134.62 

160.02 

185.42 

210.82 

236.22 

261.62 

Cm. 

10.16 

35.56 

60.96 

86.36 

111.76 

137.16 

162.56 

187.96 

213.36 

238.76 

264.16 

Cm. 

12.70 

38.10 

63.50 

88.90 

114.30 

139.70 

165.10 

190.50 

215.90 

241.30 

266.70 

Cm. 

15.24 

40.64 

66.04 

91.44 

116.84 

142.^4 

167.64 

193.04 

218.44 

243.84 

269.24 

Cm. 

17.78 

43.18 

68.58 

93.98 

119.38 

144.78 

170.18 

195.58 

220.98 

246.38 

271.78 

Cm. 

20.32 

45.72 

71.12 

96.52 

121.92 

147.32 

172.72 

198.12 

223.52 

248.92 

274.32 

Cm. 

22.86 

48.26 

73.66 

99.06 

124.46 

149.  f>6 

175.96 

200.  Si6 

226.  (« 

251.46 

276.  J^e 

CONVERSION 

OF  CENTIMETRES 

INTO  ENGLISH  INCHES 

. 

Cm. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

0 

0.000 

0.394 

0.787     1.181 

1.575 

1.969 

2.362 

2.756 

3.150 

3.543 

10 

3.937 

4.331 

4.742     5.118 

5.512 

5.906i  6.299 

6.693 

7.087 

7.480 

20 

7.874 

8.268 

8.662      9.055 

9.449 

9.843110.236  10.630 

11.024 

11.41S 

30 

11.811 

12.205    12.599    12.992 

13.386 

13.780:14.173  14.567 

14.961 

15.355 

40 

15.748 

16.142    16.530    16.929 

17.323 

17.71718.111  18.504 

18.898 

19.292 

50 

19.685 

20.079 

20.473    20.867 

21.260 

21.654:22.048  22.441 

22.835 

23.229 

60 

23.622 

24.016 

24.410    24.804 

25.197 

25  591 

25.985  26.378 

26.772 

27.166 

70 

27.560 

27.953 

28.347    28.741 

29.134 

29.528 

29.922  30.316 

30.709 

31.103 

80 

31.497 

31.890:  32.2S4    32.678 

33.071 

33.465 

33.859  34.253 

34.646 

35.040 

90 

35.434 

35.827|  36.221    36.615 

37.009 

37.402  37.796  38.190 

38.583 

38.977 

100 

39.370 

39.764    40.158    40.552 

40.945 

41. 339141. 733  42.126 

42.520 

42.914 

CONVERSION  OF  ENGLISH  FEET  INTO  METRES. 


Feet. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

0 

0.000 

0.3048 

0.'6096 

0.9144 

1.2192 

1.5239 

1.8287,2.1335 

2.4383 

2.7431 

10 

3.0479 

3.3527 

3.6575 

3.9623 

4.2671 

4.5719 

4.87675.1815 

5.4863 

5.7911 

20 

6.09.59 

6.4006 

6 . 7055 

7.0102 

7.3150 

7.6198 

7.9246  8.2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.7534 

10.058 

10.363 

10.668 

10.972  11.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.411 

13.716 

14.020  14.325 

14.630 

14.935 

50 

15.239 

15.544 

15.849 

16.154 

16.459 

16.763 

17.068  17.373 

17.678 

17.983 

60 

18.287 

18.592 

18.897 

19  202 

19.507 

19.811 

20.116  20.421 

20.726 

21.031 

70 

21.335 

21.640 

21.945 

22.250 

22.555 

22.859 

23.164  23.469 

23.774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25.907 

26.212  26.517 

26.822 

27.126 

90 

27.431 

27.736 

28.041 

28.346 

28.651 

28.955 

29.260  29.565 

29.870 

30.174 

100 

30.479 

30.784 

31.089 

31.394 

31.698 

32.003 

32.308  32.613 

32.918 

33.222 

CONVERSION  OF  METRES  INTO  ENGLISH   FEET. 


Met. 


0 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 


0 


1 


Feet. 
0.000 
32.809 
65.618 
98.4271 
131.24 
164.04 
196.85] 
229.66 
262.47 
295.28 
328.09 


Feet. 

3.2809 

36.090 

68.899 

101.71 

134.52 

167.33 

200.13 

232.94 

265.75 

298.56 

331.37 


Feet.  I 

6.5618 

39.371 

72.179 

104.99 

137.80 

170  61 

203.42 

236.22 

269.03 

391.84 

334.651 


Feet. 

9.8437 

42.651 

75.461 

108.27 

141.08 

173.89 

206.70 

239.51 

272.31 

305.12 

337.93 


Feet. 

13.123 

45.932 

78.741 

111.55 

144.36 

177.17 

209.98 

242.79 

275.60 

308.40 

341.21 


Feet. 

16.404 

49.213 

82.022 

114.83 

147.64 

180.45 

213.26 

246.07 

278.88 

311.69 

344.49, 


Feet. 
19.685 
52.494 
85.303 
118.11 
150.92 
183.73 
216.54 
249.35 
282.16 
314.97 
347.78 


Feet. 
22.966 
55.775 
88.584 
121.39 
154.20 
187.01 
219.82 
252.63 
285.44 
318.25 
351.06 


8 


Feet. 
26.247 
59.0.56 
91.865 
124.67 
157.48 
190.29 
223.10 
255.91 


321.53 
! 354. 34 


9 


Feet. 

29.528 
62.337 
95.146 
127.96 
160.76 
193. 57 
226.38 
259.19 
292.00 
324.81 
357.152 


TABLE  XXV. 


380 


CONVERSION  OF  ENGLISH  STATUTE-MILES  INTO  KILOMETRES. 


Miles. 


0 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 


0 


Kilo. 
0.0000 
16.093 
32.186 
48.2T9 
64.372 
80.46.5 
96  558 
112.65 
128.74 
144.85 
160.93 


Kilo. 

1.6093 

17.702 

33.795 

49.888 

65.981 

82.074 

98.167 

114.26 

130.35 

146.44 

16«.53 


3 


Kilo. 
3.2186 
19.312 
35.405 
51.498 
67.591 
83.684 
99  777 
115.87 
131.96 
148.05 
164  14 


Kilo. 
4.8279 
20.921 
37.014 
53.107 
69.200 
85.293 
101.39 
117.48 
133.57 
149.66 
165  75 


Kilo 

6.437 

22.530 

38.623 

54.716 

70.809 

86.902 

102.99 

119.08 

135.17 

151.26 

167.35 


Kilo. 

8.0465 

24.139 

40.232 

56.325 

72.418 

88.511 

104.60 

120.69 

136.78 

152.87 

168.96 


Kilo. 

9.6558 
25  749 
41.842 
57.935 
74.028 
90.121 
106.21 
122.30 
138.39 
154.48 
170.57 


8 


Kilo. 
11.2652  1 

27.358t 

43  451 

59.544 

75.637 

91  730 

107.82; 

123.91 i 

140.001 

156.09 

172.181 


Kilo. 

12.8745 
28.967! 
45.060! 
61.153 
i7  246! 
93.339 
109.43 
125.52 
141.61 
157.70 
173.79 


Kilo. 
14.4818 
30.577 
46,670 
62.763 
78.856 
94.949 
ill. 04 
127  13 
143.22 
159  31 
175.40 


CONVERSION  OF  KILOMETRES  INTO  ENGLISH   STATUTE-MILES. 


Kilom. 


0 
10 

20 
30 
40 
50 
60 
70 
80 
90 
100 


0 


Miles. 
0.0000 
6.2138 
12.427 
18.641 
24.8.55 
31.069 
37.282 
43.497 
49.711 
55.924 
62.138 


Miles. 
0.6214 
6.8352 
13.049 
19.263 
25.477 
31.690 
37.904 
44.118 
50.332 
56.545 
62.759 


Miles. 
1.2427 
7.4565 
13.670 
19.884 
26.098 
32.311 
38.525 
44.739 
50.953 
57.166 
,63.380 


Miles. 
1.8641 
8.0780 
14.292 
20.506 
26.720 
32.933 
39.147 
45.361 
51 . 575 
57.788 
64.002 


Miles. 
2.4855 
8.6994 
14.913 
21.127 
27.341 
33.554 
39.768 
45.982 
52.196 
58.409 
64.623 


Miles. 

3.1069 

9.3208 

15.534 

21.748 

27.962 

34.175 

40.389 

46.603 

52.817 

59.030 

65.244 


Miles. 

3.7282 

9.9421 

16.156 

23.370 

28.584 

34.797 

41.011 

47,225 

53.439 

59.652 

65.866 


Miles. 

4.3497 

10.562 

16.776 

22.990 

29.204 

35.417 

41.631 

47.845 

54.059 

60.272 

66.486 


8 


Miles. 

4.9711 

11.185 

17.399 

23.613 

29.827 

36.040 

42.254 

48.468 

54.682 

60.895 

67.109 


Miles. 

5.5924 

11.805 

18.019 

24.233 

30.447 

36.660 

42.874 

49.088 

55.302 

61.515 

67.729 


LENGTH  IN  FEET  OF  1' 


TABLE  XXVI. 

ARCS  OF  LATITUDE  AND  LONGITUDE. 


Lat. 

1'  Lat. 

r  Long. 

Lat. 

V  Lat. 

y  Long. 

1° 

6045 

6085 

31^ 

6061 

5222 

2° 

6045 

6083 

32° 

6062 

5166 

3° 

6045 

6078 

33° 

6063 

5109 

4» 

6045 

6071 

34° 

6064 

5051 

5° 

6045 

6063 

35» 

6065 

4991 

6° 

6045 

6053 

36» 

6066 

4930 

7° 

6046 

6041 

37° 

6067 

4867 

8° 

6046 

6027 

38° 

6068 

4802 

9» 

6046 

6012 

39° 

6070 

4736 

10° 

6047 

5994 

40° 

6071 

4669 

11° 

6047 

5975 

41° 

6072 

4600 

12° 

6048 

5954 

42° 

6073 

4530 

13° 

6048 

5931 

43° 

6074 

4458 

14° 

6049 

5907 

44° 

6075 

4385 

15° 

6049 

5880 

45° 

6076 

4311 

16° 

6050 

5852 

46° 

6077 

4235 

17° 

6050 

5822 

47° 

6078 

4158 

18° 

6051 

5790 

48° 

6079 

4080 

19° 

6052 

5757 

49° 

6080 

4001 

20° 

6052 

5721 

50° 

6081 

3920 

21° 

6053 

5684 

51° 

6082 

3838 

22° 

6054 

5646 

52" 

6084 

3755 

23° 

6054 

5605 

53° 

6085 

3671 

24° 

6055 

5563 

54° 

6086 

3586 

25° 

6056 

5519 

•  55° 

6087 

3499 

26° 

6057 

5474 

56° 

6088 

3413 

27° 

6058 

5427 

57° 

0089 

3323 

28° 

6059 

5378 

58° 

6090 

3233 

29° 

6060 

5327 

59° 

6091 

3142 

30° 

6061 

5275 

60° 

6092 

3051 

390 


tRIGONOMETRlC    FORMULAS. 


TABLE  XXVII. -TKIGONOMETRIC  AND   MISCELLANEOUS 

FORMULAS. 

TRIGONOMETRIC    FORMULAS. 


In  Fig  99,  let  DCE  be  the  arc  of  a  quadrant,  ABC  o.  right 
triangle,  the  angle  BA  C  subtended  by  the  arc  CE  =  A ,  and 
consider  the  radius  .1  ( '  =  unity.     Then 


BC  =sin^. 
AB  =cos^. 
H^=-tan^. 
DF  =  QotA. 
AH=s,ecA. 


AF=co^ecA. 
BE  ^=\eis.m.A. 
Zfl  =  coversin-4. 
CH=exsecA. 
CF  =coexsec  J.. 


Using  the  small  letters  a,  h,  c,  to  represent  the  sides  of  a 
right  triangle  in  Fig.  98  or  99,  v;e  may  Avrite 


sin^ 


cos^ 


a 


V 


cosec  A 


.-.  sin^ 


a 


cosec  A 
1 


sec-4^-;   .-.  q.osA:= 

c  sec  A 


tan-4  =  -;       cot^=-;  .-.  tan^  ^ 

c  a  cotA 


N 


SOLUTION   OF   TRIANGLES. 


301 


TABLE  XXVII.— TRIGONOMETRIC  AND   MISCELLANEOUS 

FORMULAS. 

SOLUTION   OF  RIGHT  TRIANGLES. 


Required. 

Given. 

A,  C,  C 

a,  b 

A,  C,  b 

a,  c 

C,  6,  c 

A,  a 

C,  a,  c 

A,  b 

C,  a,  b 

A,  c 

Formulas. 


sin  A  =  cos  C  =  - ;    c  =  ^{b  +  a)  {b  —  a)^ 


tSinA  =cotB 


a 


b  =  Va-^  +  c2. 


C  ^^  90°  —  A  ;  c  =  a  cot  A  ;  b  =  a  cosec  vl. 
C  =  90°  —  A  ;  a  =  bsmA  ;  c  =  b cosin  A, 
C  =  90°  —  A;  a  =  ctan^;  6  =  csec^. 


SOLUTION   OF  OBLIQUE  TRIANGLES. 


Required. 

b 

B 

1,{A  +  B) 
h{A  -  B) 

A 
B 


Area 
Area 

Area 


Given. 
A,  B,  a 

A,  a,  b 


\  a.b,  C 


<> 


a,  6,  c 


> 


< 


A^  &,  c 
A,  B,  c 


Formulas. 


b  = 


a  sin  B 
sin -4 

bsinA 


sin  B  - 

i{A  +  B)  =  ^{m-C)  /^z^^^^^'^'^ 

tani(^  -  B)  =^^^tan'iM  +  B) 
a  +  6    • 

A--=^i(A  +  B)  +  i(A-B) 

B=.i(A  +  B)-i{A-  B) 

If  s=i{a+b-\-c),  smiA=^l^^~^l^^~'') 

\  be 


\      s  (s  —  a) 

-^inl       2x/s(8-a)(s-6)(s- 

-c) 

be 

Area  =  -v  s  (s  —  a){s  —  b)  (s  —  c) 
Area  =  I  be  sin  A 

c^  sin  ^  sin  B 


Area  = 


2sin(^+  jB) 


•^•^J'-  GEXERAL    FORMULAS. 


TABLE  XXVII.-TRIGONOMETRIC  AND   MISCELLANEOUS 

FORMULAS. 

GENERAL  FORMULAS. 

sin^  =  V 1  —  cos^^  =  tan^  cos  A. 
sin  A  =2  sin  ^-4  cos  ^-4. 

sin  A  = ^ =  V'i(l  —  C0S24). 

cosec-4 

cos^  = =  V^l  —  sin2^  =  cot^  sin^- 

secA 

cosA  =1—2  sin2^^  ^  1  —  vers^. 

cosA  =  ^i  +  icos2^  =  cos2^^  —  sin2^^. 

tan^  =  ^^  =  Vsec'^A  -  1. 
cos  A 


tan -4 


cos^  1  +  cos2yl 

1  1— cos2^ 


cot^  sin  2^ 


^^+  A  1  cos -4         / T-z rr 

cot  A  = = =  V  cosec^^  —  1. 

tan  A       sin  A 

cot^  =  _5HlM_ = LL^^^M. 

1  —  cos2^  sin  2^ 

sec -4  = =  the  reciprocal  of  any  expression  for  cos^. 

cos  J. 


cosec  A 

1 

=  the 

sin 

A 

vers^ 

—  1  - 

-  cosA  — 

2sinH^. 

exsec  A 

—  sec 

A- 

-1  — 

vers^ 

cos^ 


Bini^  =  JIUSS^  =     /ZH 


s^ 


G ENTERAL    FORMULAS.  393 

TABLE  XXVII. -TRIGONOMETRIC  AND   MISCELLANEOUS 

FORMULAS. 


2 
tan  A  1  —  cos^  sin -4 


coiiA  = 


1  +  sec^  siii^  1  +  cos  A 

1  +  cos^  sinA 


sin  A  1  —  cos  J. 

sin  2  A  =2  sin  A  cos  ^. 
cos2^  ==  cos2  J.  —  sin2  J.  =  2  cos^^  —  L 

2tan^ 


tan  2^ 


cot  2^ 


1  —  tan2^ 
cot2^  —  1 


2cot^ 

sin  {A  ±  7^)  ^  sin  ^  cos  B  ±  cos  A  sin  B, 
cos(yl  ±  7^)  =  cosyl  cos5  ip  sin  J.  sinJ?. 

tan(^  ±  B)  =  tan^±tanJ?^ 
1  q=  tan  A  tan  B 

&m.A  +  sin  J?  =  2  sin  |(  J.  +  i?)  cos|(J.  —  B). 

sin  A  —  sin  Ji  =  2  cos  i{A  +  B)  sin  i  (^  —  5). 

cos^  +  cos 7?  =2 cos i (J.  +  B)  cosi{A  —  B). 

cosB  — cos^  =  2sini(^  +  5)sini(^  —  B). 

sin2^  —  sin2  5  =  cos2  5  —  cosS^l  =  sin  (J.  +  B)  Gin(^  —  B). 

cos2^  -  sin2J5  =  cos  {A  +  B)  cos  {A  —  B). 

tan^±tan£  =  ^HiiA±JZ). 
cos^  cos^ 

cotA±o.oiB^±^''^^M, 

6iaA  '6m.B 


391 


MISCELLANEOUS   FORMULAS. 


TABLE  XXVII. —TRIGONOMETRIC  AND   MISCELLANEOUS 

FORMULAS. 

MISCELLANEOUS   FORMULAS. 


Required. 

Area  of 
Trapezoid 

Regular  Polygon 

Circle 

Ellipse 

Parabola 

Surface  of 
Cone 

Cylinder 
Sphere 

Zone 

Voliune  of 
Prism  or  cylinder 

l^yraniid  or  cone 

Frustum  of 

Pyramid  or  cone 

Sphere 


Given. 

Parallel  sides  =  m  and  n 
Perp.  dist.  bet.  them  =p 
Length  of  side  =  I 
Number  of  sides  =  n 
Radius  =  r 
Semi-axes  =  a  and  b 
Base  =  6,  height  =  h 

Radius  of  base  =  r 

Slant  height  =  s 

Radius  =  r,  height  =  h 

Radius  =  r 

Height  =  h 

Radius  of  its  sphere  =  r 


Area  of  base 
Height  =  h 
Area  of  base 
Height  =  h 


h 
h 


Area  of  bases 
h 
r 


b  and  b' 


Height : 
Radius 


Formulas. 


P 

2 


(m  +  n) 


—  cot • 

4  n 

7tr^  [;r  =  3.1416] 

Ttab 

Ibh 


Ttrs 


2  7trh 
4  7cr^ 

2  7trh 


bh 

bh 
3 


-  (b  +V  +  ^/bb' 
3 


|;rr3 


TABLE   XXVIII. — SQUARE   AND   CUBE   ROOTS.        305 


Sq 

uare  Roots  and  Cube  Roots  of  Numbers  from  .1  to  28. 

Ko 

errors. 

No. 

Square. 

Cube. 

Sq.  Bt. 

C.  Rt. 

No. 

9q.  at. 

C.  Et. 

No. 

Sq.  Rt. 

C.  Rt. 

.1 

.01 

.001 

.316 

.464 

.7 

2.387 

1.786 

.4 

3.661 

2.375 

.15 

.0225 

.0034 

.387 

.531 

.8 

2.408 

1.797 

.6 

3.688 

2.387 

.2 

.04 

.008 

.447 

.585 

.9 

2.429 

1.807 

.8 

3.715 

2.399 

.25 

.0625 

.0156 

.500 

.630 

6. 

2.449 

1.817 

14. 

3.742 

2.410 

.3 

.09 

.027 

.548 

.669 

.1 

2.470 

1.827 

.2 

3.768 

2.422 

.35 

.1225 

.0429 

.592 

.705 

.2 

2.490 

1.837 

.4 

3.795 

2.433 

.4 

.16 

.064 

.633 

.737 

.3 

2.510 

1.847 

.6 

3.821 

2.44A 

.45 

.2025 

.0911 

.671 

.766 

.4 

2.530 

1.857 

.8 

3.847 

2,455 

.5 

.25 

.125 

.707 

.794 

.5 

2.550 

1.866 

15. 

3.873 

2.466 

.85 

.3025 

.1664 

.742 

.819 

.6 

2.569 

1.876 

.2 

3.899 

2.477 

.6 

.36 

.216 

.775 

.843 

.7 

2.588 

1.885 

.4 

3.924 

2.488 

.65 

.4225 

.2746 

.806 

.866 

.8 

2.608 

1.895 

.6 

3.950 

2.499 

.7 

.49 

.343 

837 

.888 

.9 

2.627 

1.904 

.8 

3.975 

2.509 

.75 

.5625 

.4219 

.866 

.909 

7. 

2.646 

1.913 

16. 

4. 

2.520 

.8 

.64 

.512 

.894 

.928 

.1 

2.665 

1.922 

.2 

4.025 

2.530 

.85 

.7225 

.6141 

.922 

.947 

.2 

2.683 

1.931 

.4 

4.050 

2.541 

.9 

.81 

.729 

.949 

.965 

.3 

2.702 

1.940 

.6 

4.074 

2.551 

» 

.9025 

.8574 

.975 

.983 

.4 

2.720 

1.949 

.8 

4.099 

2.561 

]. 

1.000 

1.000 

1.000 

1.000 

.5 

2.739 

1.957 

17. 

4.123 

2.571 

.05 

1.103 

1.158 

1.025 

1.016 

.6 

2.757 

1.966 

.2 

4.147 

2.581 

).l 

1.210 

1.331 

1.049 

1.032 

.7 

2.775 

1.975 

.4 

4.171 

2.591 

.15 

1.323 

1.521 

1.072 

1.048 

.8 

2.793 

1.983 

.6 

4.195 

2.601 

1.2 

1.440 

1.728 

1.095 

1.063 

.9 

2.811 

1.992 

.8 

4.219 

2.611 

.25 

1.563 

1.963 

1.118 

1.077 

8. 

2.828 

2.000 

18. 

4.243 

2.621 

5.3 

1.690 

2.197 

1.140 

1.091 

.1 

2.846 

2.008 

.2 

4.266 

2.630 

.35 

1.823 

2.460 

1.162 

1.105 

.2 

2.864 

2.017 

.4 

4.290 

2.640 

1.4 

1.960 

2.744 

1.183 

1.119 

.3 

2.881 

2.025 

.6 

4.313 

2.650 

.45 

2.103 

3.049 

1.204 

1.132 

.4 

2.898 

2.033 

.8 

4.336 

2.659 

1.5 

2.250 

3.375 

1.225 

1.145 

.5 

2.915 

2.041 

19. 

4.359 

2.668 

.55 

2.403 

3.724 

1.245 

1.157 

.6 

2.93S 

2.049 

.2 

4.382 

2.67tt 

1.6 

2.560 

4.096 

1.265 

1.170 

.7 

2.950 

2.057 

.4 

4.405 

2.687 

,65 

2.723 

4.492 

1.285 

1.182 

.8 

2.966 

2.065 

.6 

4.427 

2.696 

).7 

2.890 

4.913 

1.304 

1.193 

.9 

2.983 

2.072 

.8 

4.450 

2.705 

.75 

3.063 

5.359 

1.323 

1.205 

9. 

3. 

2.080 

20. 

4.472 

2.714 

1.8 

3.240 

5.832 

1.342 

1.216 

.1 

3.017 

2.088 

.2 

4.494 

2.723 

.85 

3.423 

6.332 

1.360 

1.228 

.2 

3.033 

2.095 

.4 

4.517 

2.732 

1,9 

3.610 

6.859 

1.378 

1.239 

.3 

3.050 

2.103 

.6 

4.539 

2.741 

.95 

3.803 

7.415 

1.396 

1.249 

.4 

3.066 

2.110 

.8 

4.561 

2.750 

til. 

4.000 

8.000 

1.414 

1.260 

.5 

3.082 

2.118 

21. 

4.583 

2.759 

.1 

4.410 

9.261 

1.449 

1.281 

.6 

3.098 

2.125 

.2 

4.604 

2.768 

.2 

4.840 

10.65 

1.483 

1.301 

.7 

3.114 

2.133 

.4 

4.626 

2.776 

.3 

5.290 

12.17 

1.517 

1.320 

.8 

3.130 

2.140 

.6 

4.648 

2.785 

.4 

5.760 

13.82 

1.549 

1.339 

.9 

3.146 

2.147 

.8 

4.669 

2.794 
2.802 

.5 

6.250 

15.63 

1.581 

1.357 

10. 

3.162 

2.154 

22. 

4.690 

.6 

6.760 

17.58 

1.612 

1.375 

.1 

3.178 

2.162 

.2 

4.712 

2.810 

.7 

7.290 

19.68 

1.643 

1.392 

.2 

3.194 

2.169 

.4 

4.733 

2.819 

.8 

7.840 

21.95 

1.673 

1.409 

.3 

3.209 

2.176 

.6 

4.754 

2.827 

.9 

8.410 

24.39 

1.703 

1.426 

.4 

3.225 

2.183 

.8 

4.775 

2.836 

8. 

9. 

27. 

1.732 

1.442 

.5 

3.240 

2.190 

23. 

4.796 

2.844 

.1 

9.61 

29.79 

1.761 

1.458 

.6 

3.256 

2.197 

.2 

4.817 

2.852 

.2 

10.24 

32.77 

1.789 

1.474 

.7 

3.271 

2.204 

.4 

4.837 

2.860 

.3 

10.89 

35.94 

1.817 

1.489 

.8 

3.286 

2.210 

.6 

4.858 

2.868 

.4 

11.56 

39.30 

1.844 

1.504 

.9 

3.302 

2.217 

.8 

4.879 

2.876 

.5 

12.25 

42.88 

1.871 

1.518 

11. 

3.317 

2.224 

24. 

4.899 

2.884 

.6 

12.96 

46.66 

1.897 

1.533 

.1 

3.332 

2.231 

.2 

4.919 

2.892 

.7 

13.69 

50.65 

1.924 

1.547 

.2 

3.347 

2.287 

.4 

4.940 

2.900 

.8 

14.44 

54.87 

1.949 

1.560 

.3 

3.362 

2.244 

.6 

4.960 

2.908 

.9 

15.21 

59.32 

1.975 

1.574 

.4 

3.376 

2.251 

.8 

4.980 

2.916 

4. 

16. 

64. 

.     2. 

1.587 

.5 

3.391 

2.257 

25. 

5. 

2.924 

.1 

16.81 

68.92 

2.025 

1.601 

.6 

3.406 

2.264 

.2 

5.020 

2.932 

.2 

17.64 

74.09 

2.049 

1.613 

.7 

3.421 

2.270 

.4 

5.040 

2.940 

.3 

18.49 

79.51 

2.074 

1.626 

.8 

3.435 

2.277 

.6 

5.060 

2.947 

.4 

19.36 

85.18 

2.098 

1.639 

.9 

3.450 

2.283 

.8 

5.079 

2.955 

.5 

20.25 

91.13 

2.121 

1.651 

12. 

3.464 

2.289 

26. 

5.099 

2.962 

.6 

21.16 

97.34 

2.145 

1.663 

.1 

3.479 

2.296 

.2 

5.119 

2.970 

.7 

22.09 

103.8 

2.168 

1.675 

.2 

3.493 

2.302 

.4 

5.138 

2.978 

.8 

23.04 

110.6 

2.191 

1.687 

.3 

3.507 

2.308 

.6 

5.158 

2.985 

.9 

24.01 

117.6 

2.214 

1.698 

.4 

3.521 

2.315 

.8 

5.177 

2.993 

,1 

25. 

125. 

2.236 

1.710 

.5 

3.536 

2.321 

27. 

5.196 

3.000 

26.01 

132.7 

2.258 

1.721 

.6 

3.550 

2.327 

.2 

5.215 

3.007 

.2 

27.04 

140.6 

2.280 

1.732 

.7 

3.564 

2.333 

.4 

5.235 

3.015 

.3 

28.09 

148.9 

2.302 

1.744 

.8 

3.578 

2.339 

.6 

5.254 

3.022 

,4 

29.16 

157.5 

2.324 

1.754 

.9 

3.592 

2.345 

.8 

5.273 

3.029 

.6 

30.25 

166.4 

2.345 

1.765 

13. 

3.606 

2..351 

28. 

5.292 

8.037 

.8 

S1.3« 

176.6 

2.366 

1.776 

.2 

3.633 

2.363 

.2 

&.310 

3.044 

S9e      TABLE    XXIX.— SQUARES,    CUBES,    AKD    ROOTS. 


TABL.E  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roets« 
of  ^''umbers  from  1  to  1000. 

Kemabk  OS  THB  poLLowiNs  Taelk.     Wherever  the  effect  of  a  fifth  decimal  in  the  roots  would  be  t* 
idd  I  to  tte  fourth  and  final  decimal  in  the  table,  the  addition  has  been  made.  No  errors. 


No. 

Square. 

Cube. 

Sq.  Kt. 

C.  Bt. 

No. 

Square 

Cube. 

Sq.  Kt. 

C.  Kt. 

1 

1 

1 

1.0000 

1.0000 

61 

3721 

226981 

7.8102 

3.9365 

2 

4 

8 

1.4142 

1.2599 

62 

3844 

238328 

7.8740 

3.9579 

3 

9 

27 

1.7321 

■  1.4422 

63 

3969 

250047 

7.9373 

3.9791 

4 

16 

64 

2.0000 

1.5874 

64 

4096 

262144 

8.0000 

4. 

5 

25 

125 

2.2361 

1.7100 

65 

4225 

274625 

8.0623 

4.0207 

e 

36 

216 

2.4495 

1.8171 

66 

4356 

287496 

8.1240 

4.0412 

7 

49 

343 

2.6458 

1.9129 

67 

4489 

300763 

8.1854 

4.0615 

8 

64 

512 

2.8284 

2.0000 

68 

4624 

314432 

6.2462 

4.0817 

8 

81 

729 

3.0000 

2.0801 

69 

4761 

328509 

8.3066 

4.1016 

M 

100 

1000 

3.1623 

2.1544 

70 

4900 

343000 

8.3666 

4.1213 

11 

121 

1331 

3.3166 

2.2240 

71 

5041 

357911 

8.4-261 

4.1408 

12 

144 

1728 

3.4641 

2.2894 

72 

5184 

373248 

8.4853 

4.1602 

13 

169 

2197 

3.6056 

2.3513 

73 

5329 

389017 

8.5440 

4.1793 

14 

196 

2744 

3.7417 

2.4101 

74 

5476 

405224 

8.6023 

4.1983 

15 

225 

3375 

3.8730 

2.4662 

75 

5625 

421875 

8.6603 

4.2173 

16 

256 

4096 

4.0000 

2.5198 

76 

5776 

438976 

8.7178 

4.2358 

17 

2S9 

4913 

4.1231 

2.5713 

77 

5929 

456533 

8.7750 

4.-2543 

18 

324 

5&32 

4.2426 

2.6207 

78 

6084 

474552 

8.8318 

4.2727 

19 

361 

6859 

4.3589 

2.6684 

79 

6241 

493039 

8.8882 

4.-2908 

•X 

400 

8000 

4.4721 

2.7144 

80 

6400 

512000 

8.9443 

4.3089 

21 

441 

9261 

4.5826 

2.7589 

81 

6561 

531441 

9. 

4.3267 

22 

484 

10648 

4.6904 

2.8020 

82 

6724 

551368 

9.0554 

4.3445 

23 

529 

12167 

4.7958 

2.8439 

83 

6889 

5<rr'87 

9.1104 

4.3621 

24 

576 

13824 

4.8990 

2.8845 

84 

7056 

592704 

9.1652 

4.3795 

25 

625 

1562.T 

5.0000 

2.9240 

85 

7225 

6141-25 

9.2195 

4.3968 

26 

676 

17576 

5.0990 

2.9625 

86 

7396 

636056 

9.2736 

4.4140 

27 

729 

1968;} 

5.1962 

3.0000 

87 

7569 

658503 

9.3274 

4.4310 

28 

784 

21952 

5.2915 

3.0.',66 

88 

7744 

681472 

9.3808 

4.4480 

29 

841 

243S9 

5.3852 

3.(J72;{ 

89 

7921 

704969 

9.4340 

4.4647 

30 

900 

27000 

5.4772 

3.1072 

90 

8100 

729000 

9.4868 

4.4814 

31 

%1 

29791 

5.5678 

3.1414 

91 

8281 

753571 

9.5394 

4.4979 

32 

1024 

32768 

5.6569 

3.1748 

92 

8464 

77S688 

9.5917 

4.5144 

33 

1089 

35937 

5.7446 

3.2075 

93 

8649 

804357 

9.6437 

4.5307 

34 

1156 

39304 

5..s;iio 

3.2.396 

94 

88,36 

830584 

9.6954 

4.54«8 

35 

1225 

42875 

5.9161 

3.2711 

95 

9025 

857375 

9.7468 

4.56-29 

36 

1296 

46656 

6.0000 

3.3019 

96 

9216 

884736 

9.7980 

4.5789 

37 

1369 

50653 

6.0828 

3.3322 

97 

9409 

912673 

9.8489 

4.59t7 

38 

1444 

54872 

6.1644 

3.. 3620 

98 

9604 

941192 

9.8995 

4.6104 

39 

1521 

59319 

6.2450 

3.3912 

99 

9801 

970299 

9.9499 

4.6261 

40 

1600 

64000 

6.3246 

3.4200 

100 

10009 

1000000 

10. 

4.6416 

41 

1681 

68921 

6.4031 

3.4482 

101 

10201 

1030301 

10.0499 

4.6570 

42 

1764 

74088 

6  4807 

3.4760 

102 

10404 

1061208 

10.0995 

4.67-23 

43 

1849 

79507  ' 

6.5574 

3.5034 

103 

10609 

1092727 

10.1489 

4.6875 

44     i 

1936 

85184 

6.6332 

3.5303 

104 

10816 

1124864 

10.1980 

4.7027 

45 

2025 

91125 

6.7082 

3.5569 

105 

110-25 

1J576'25 

10.2470 

4.7177 

46 

2116 

97336 

6.7823 

3.5830 

106 

11236 

1191016 

10.2956 

4.7326 

47     / 

2209 

103823 

6.8557 

3.60'-8 

107 

11449 

1225043 

10.3441 

4.7475 

48     1 

2304 

110593 

6.9282 

3.6342 

108 

11664 

1-259712 

10.3923 

4.7622 

49 

2401 

117649 

7.0000 

3.6593 

109 

11881 

1295029 

10.4403 

4.7769 

50 

2500 

125000 

7.0711 

3.6840 

110 

12100 

1331000 

10.4881 

4.7914 

SI 

2601 

132651 

7.1414 

3.7084 

111 

12321 

1367631 

10.5357 

4.8059 

52 

2704 

140608 

7.2111 

3.7325 

112 

12544 

14049-28 

10.5830 

4.8203 

53 

2809 

148877 

7.2801 

3.756:} 

113 

12769 

144-2897 

10.6301 

4.8346 

54 

2916 

157464 

7.3485 

3.7798 

114 

1-2996 

1481544 

10.6771 

4.8488 

55 

3025 

166375 

7.4162 

3.8030 

115 

13225 

1520875 

10.7238 

4.8629 

56 

3136 

175616 

7.4833 

3.8259 

116 

13456 

1560896 

10.7703 

4.8770 

57 

3249 

185193 

7.5498 

3.8485 

117 

1.3689 

1601613 

10.8167 

4.8910 

58 

3364 

■  195112 

7.6158 

3.8709 

118 

139-24 

1643032 

10.8628 

4.9048 

59 

34«1 

205379 

7.6811 

3.8930 

119 

14161 

1685159 

10.9087 

4.9187 

CO 

3600 

216000 

T.7460 

3.9149 

120 

14400 

1728000 

10.9515 

4.93M 

TABLE    XXTX. — St^UARES,    CUBES,    AND    ROOTS.     39? 


TABIjE  of  Sqnaros.  C'libes.  Square  Roots,  an<l  €nbe  Roots, 
of  Numbers  from  1  to  1000  —  (Continued.) 


No. 

Square. 

Cube. 

Sq.  Kt. 

cut. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

121 

14641 

1771561 

11. 

4.9461 

186 

34596 

6434856 

13.6382 

5.7083 

122 

14884 

1815848 

11.0454 

4.9597 

187 

34969 

6539203 

13.6748 

5.7185 

123 

15129 

1860867 

11.0905 

4.9732 

188 

35344 

6644672 

13.7113 

5.7287 

124 

15376 

190H624 

11.1355 

4.9866 

189 

35721 

6751269 

13.7477 

5.7388 

125 

15625 

1953125 

11.1803 

5. 

190 

36100 

6859000 

13.7840 

5.7489 

126 

15876 

2000376 

11.2250 

5.0133 

191 

36481 

6967871 

13.8203 

5.7590 

127 

16129 

2048383 

11.2694 

5.0265 

192 

36864 

7077888 

13.8564 

5.7690 

128 

16384 

2097152 

11.3137 

5.0397 

193 

37249 

7189057 

13.8924 

5.7790 

129 

16641 

2146689 

11.3578 

5.0528 

194 

37636 

7301 o84 

13.9284 

5.7890 

130 

16900 

2197000 

11.4018 

5.0658 

195 

38025 

7414875 

13.9642 

5.7989 

131 

17161 

2248091 

11.4455 

5.0788 

196 

38416 

7529536 

14. 

5.8088 

132 

17424 

2299968 

11.4891 

5.0916 

197 

38809 

7645373 

14.0357 

5.8186 

133 

17689 

2352637 

11.5326 

5.1045 

198 

39204 

7762392 

14.0712 

5.8285 

134 

17956 

2406104 

11.5758 

5.1172 

199 

39601 

7880599 

14.1067 

5.8383 

135 

18225 

2460375 

11.6190 

5.1299 

200 

40000 

8000000 

14.1421 

5.8480 

136 

18496 

2515456 

11.6619 

5.1426 

201 

40401 

8120601 

14.1774 

5.8578 

137 

18769 

2571353 

11.7017 

5.1551 

202 

40804 

8242408 

14.2127 

5.8675 

138 

19044 

2628072 

11.7473 

5.1676 

203 

41209 

8365427 

14.2478 

5.8771 

139 

19321 

2685619 

11.7898 

5.1801 

204 

41616 

8489664 

14.2829 

5.8869 

110 

19600 

2744000 

11.8322 

5.1925 

205 

42025 

8615125 

14.3178 

5.8964 

141 

19881 

2803221 

11.8743 

5.2048 

206 

42436 

8741816 

14.3527 

5.9059 

142 

20164 

2863288 

11.9164 

5.2171 

207 

42849 

8869743 

14.3875 

5.9155 

143 

20449 

2924207 

11.9583 

5.2293 

208 

43264 

8998912 

14.4222 

5.9250 

144 

20736 

2985984 

12. 

5.2415 

209 

43681 

9129329 

14.4568 

5.9345 

145 

21025 

3048625 

12.0416 

5.2536 

210 

44100 

9261000 

14.4914 

5.9439 

146 

21316 

3112136 

12.0830 

5.2656 

211 

44521 

9393931 

14.5258 

5.9533 

147 

21609 

3176523  . 

12.1244 

5.2776 

212 

44944 

9528128 

14.5602 

5.9627 

148 

21904 

3241792 

12.1655 

5.2896 

213 

45369 

9663597 

14.5945 

5.9721 

149 

22201 

3307949 

12.2066 

5.3015 

214 

•  45796 

9800344 

14.6287 

5.9814 

150 

22500 

3375000 

12.2474 

5.3133 

215 

46225 

9938375 

14.6629 

5.9907 

151 

22801 

3442951 

12.2882 

5.3251 

216 

46656 

10077696 

14.6969 

6. 

152 

23104 

3511808 

12.3288 

5.3368 

217 

47089 

10218313 

14.7309 

6.0092 

153 

23409 

3581577 

12.3693 

5.3485 

218 

47524 

10360232 

14.7648 

6.0185 

154 

23716 

3652264 

12.4097 

5.3601 

219 

47961 

10503459 

14.7986 

6.0277 

155 

24025 

3723875 

12.4499 

5.3717 

220 

48400 

10648000 

14.8324 

6.0368 

156: 

24336 

3796416 

12.4900 

5.3832 

221 

48841 

10793861 

14.8661 

6.0459 

157 

24649 

3869893 

12.5300 

5.3947 

222 

49284 

10941048 

14.8997 

6.0550 

158 

24964 

3944312 

12.5698 

5.4061 

223 

49729 

11089567 

14.9332 

6.0641 

159 

25281 

4019679 

12.6095 

5.4175 

224 

50176 

11239424 

14.9666 

6.0732 

160 

25600 

4096000 

12.6491 

5.4288 

225 

50625 

11390625 

15. 

6.0822 

161 

25921 

4173281 

12.6886 

5.4401 

226 

51076 

11543176 

15.0333 

6.0912 

162 

26244 

4251528 

12.7279 

5.4514 

227 

51529 

11697083 

15.0665 

6.1002 

163 

26569 

4330747 

12.7671 

5.4626 

228 

51984 

11852352 

15.0997 

6.1091 

164 

26896 

4410944 

12.8062 

5.4737 

229 

52441 

12008989 

15.1327 

6.1180 

165 

27225 

4492125 

12.8462 

5.4848 

230 

52900 

12167000 

15.1658 

6.1269 

166 

27550 

4574296 

12.8841 

5.4959 

231 

53361 

12326391 

15.1987 

6.1358 

167 

27889 

4657463 

12.9228 

5.5069 

232 

53824 

12487168 

15.2315 

6.1446 

168 

28224 

4741632 

12.9615 

5.5178 

233 

54289 

12649337 

15.2643 

6.1534 

169 

28561 

4826809 

13. 

5.5288 

234 

54756 

12812904 

15.2971 

6.1622 

170 

28900 

4913000 

13.0384 

5.5397 

235 

55225 

12977875 

15.3297 

6.1710 

171 

29241 

5000211 

13.0767 

5.5505 

236 

55696 

13144256 

15.3623 

6.1797 

172 

29584 

5088448 

13.1149 

5.5613 

237 

56169 

13312053 

.15.3948 

6.1885 

173 

29929 

5177717 

13.1.529 

5.5721 

238 

56644 

13481272 

15.4272 

6.1972 

174 

30276 

5268024 

13.1909 

5.5828 

239 

57121 

1'3651919 

15.4596 

6.2058 

175 

30625 

5359375 

13.2288 

5.5934 

240 

57600 

13824000 

15.4919 

6.2145 

176 

30976 

5451776 

13.2665 

5.6041 

241 

58081 

13997521 

15.5242 

6.2231 

177 

31329 

5545233 

13.3041 

5.6147 

242 

58564 

14172488 

15.5563 

6.2317 

178 

31684 

5639752 

13..3417 

5.6252 

243 

59049 

14348907 

15.5885 

6.2403 

179 

32041 

5735339 

13.3791 

5.6357 

244 

59536 

14526784 

15.6205 

6.2488 

180 

32400 

5832000 

13.4164 

5.6462 

245 

60025 

14706125 

15.6525 

6.2573 

181 

32761 

5929741 

13.4536 

5.6567 

246 

60516 

14886936 

15.6844 

6.2658 

182 

33124 

6028568 

13.4907 

5.6671 
5.6V74 

247 

61009 

15069228 

15.7162 

6.2743 

183 

33489 

6128487 

13.5277 

248 

61504 

15252992 

15.7480 

6.2828 

184 

33856 

6229504 

13.5647 

5.6877 

249 

62001 

1543824« 

15.7797 

6.2912 

186 

84225 

6S31625 

13.6015 

5.6980 

250 

62500 

15625000 

15.8114 

6.29M 

398     TAELE    XXIX.  —  SQUARES,    CUBES,    AND    ROOTS. 


TABIiE  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1000  —  (Continued.) 


Ko. 

Square. 

Cube. 

Sq.  Rt. 

C.  Bt. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt, 

351 

63001 

15813251 

15.8430 

6..3080 

316 

99856 

31554496 

17.7764 

6.8113 

252 

63504 

16003008 

15.8745 

6.3164 

317 

100489 

31855013 

17.8045 

6.8185 

353 

64009 

16194277 

15.9060 

6.3247 

318 

101124 

32157432 

17.8326 

6.8256 

354 

64516 

16387064 

15.9374 

6..S330 

319 

101761 

32461759 

17.8606 

S.8328 

255 

65025 

16581375 

15.9687 

6.3413 

320 

102400 

32768000 

17.8885 

6.8399 

256 

65536 

1677T216 

16. 

6.3496 

321 

103041 

33076161 

17.9165 

6.8470 

257 

66049 

16974593 

16.0312 

6.3579 

322 

103684 

33386248 

17.9444 

6.8541 

258 

66564 

17173512 

16.0624 

6.3661 

323 

104329 

33698267 

17.9722 

6.8612 

259 

67081 

17373979 

16.0935 

6.3743 

324 

104976 

34012224 

18. 

6.8683 

260 

67600 

17576000 

16.1245 

6.3825 

325 

105625 

34328125 

18.0278 

6.8753 

261 

68121 

17779581 

16.1555 

6..3907 

326 

106276 

34645976 

18.0555 

6.8824 

262 

68644 

17984728 

16.1864 

6.3988 

327 

106929 

34965783 

18.0831 

6.8894 

263 

69169 

18191447 

16.2173 

6.4070 

328 

107584 

35287552 

18.1108 

6.8964 

264 

69696 

18399744 

16.2481 

6.4151 

329 

108241 

35611289 

18.1384 

6.9034 

265 

70225 

18609625 

16.2788 

6.4232 

330 

108900 

35937000 

18.1659 

6.91W 

266 

70756 

18821096 

16.3095 

6.4312 

331 

109561 

36264691 

18.1934 

6.9171 

267 

71289 

19034163 

16.3401 

6.4393 

332 

110224 

36594368 

18.2209 

6.9244 

368 

71824 

19248832 

16.3707 

6.4473 

3.33 

110889 

36926037 

18.2483 

6.9313 

269 

72361 

19465109 

16.4012 

6.4553 

334 

111556 

37259704 

18.2757 

6. 9383 

270 

72900 

19683000 

16.4317 

6.4633 

335 

112225 

37595375 

18.3030 

6.9457 

271 

73441 

19902511 

16.4621 

6.4713 

3.36 

112896 

37933056 

18.3303 

6.9521 

272 

73984 

20123648 

16.4924 

6.4792 

337 

113569 

38272753 

18.3576 

6.9589 

273 

74529 

20346417 

16.5227 

6.4872 

338 

114244 

38614472 

18.3848 

6.9658 

274 

75076 

20570824 

16.5529 

6.4951 

339 

114921 

38958219 

18.4120 

6.9727 

275 

75625 

20796875 

16.5831 

6.5030 

340 

115600 

39304000 

18.4391 

6.979f 

276 

76176 

21024576 

16.6132 

6.5108 

341 

116281 

39651821 

18.4662 

6.9864 

277 

76729 

21253933 

16.6433 

6.5187 

342 

116964 

40001688 

18.4932 

6.993) 

278 

77284 

21484952 

16.6733 

6.5265 

343 

117649 

40353607 

18.5203 

7. 

279 

77841 

21717639 

16.7033 

6..5343 

344 

na336 

40707584 

18.5472 

7.006J 

280 

78400 

21952000 

16.7332 

6.5421 

345 

119025 

41063625 

18.5742 

7.013* 

281 

78961 

22188041 

16.7631 

6.5499 

346 

119716 

41421736 

18.6011 

7.0203 

282 

79524 

22425768 

16.7929 

6..5577 

.347 

120409 

41781923 

18.6279 

7.0271 

383 

80089 

22665187 

16.8226 

6.5654 

348 

121104 

42144192 

18.6548 

7.0338 

284 

80656 

22906304 

16.8523 

6.5731 

349 

121801 

42508549 

18.6815 

7.040'.- 

285 

81225 

23149125 

16.8819 

S.5808 

350 

122500 

42875000 

18.7083 

7.047  I 

286 

81796 

23393656 

16.9115 

6.5885 

.351 

123201 

4324.3551 

18.7.350 

7.0540 

287 

82369 

23639903 

16.9411 

6.5962 

352 

123904 

43614208 

18.7617 

7.0607 

288 

82944 

23887872 

16.9706 

6.60.39 

353 

124609 

43986977 

18.7883 

7.0674 

289 

83521 

24137569 

17. 

6.6115 

354 

125316 

44361864 

18.8149 

7.0740 

290 

84100 

24389000 

17.0294 

6.6191 

355 

1  •260-25 

44738875 

18.8414 

7.0807 

391 

84681 

24642171 

17.0587 

6.6267 

356 

1267.36 

45118016 

18.8680 

7.0873 

292 

85264 

24897088 

17.0880 

6.6343 

.357 

127449 

45499293 

18.8944 

7.0940 

393 

85849 

25153757 

17.1172 

6.6419 

358 

128164 

45882712 

18.9209 

7.1006 

394 

86436 

25412184 

17.1464 

6.6494 

359 

128881 

46268279 

18.9473 

7.1072 

395 

87025 

2567-2375 

17.1756 

6.6569 

360 

129600 

46656000 

18.9737 

7.1138 

296 

87616 

25934336 

17.2047 

6.6644 

361 

1.30321 

47045881 

19. 

7.1204 

297 

88209 

26198073 

17.23.37 

6.6719 

362 

1.31044 

47437928 

19.0263 

7.1269 

298 

88804 

26463592 

17.2627 

6.6794 

363 

131769 

47832147 

19.0526 

7.1335 

399 

89401 

26730899 

17.2916 

6.6869 

364 

132496 

48228544 

19.0788 

7.1400 

300 

90000 

27000000 

17.3205 

6.6943 

365 

133225 

48627125 

19.1050 

7.1466 

301 

90601 

27270901 

17.3494 

6.7018 

366 

1.3.3956 

49027896 

19.1311 

7.15.31 

302 

91204 

27543608 

17.3781 

6.7092 

367 

134689 

494;30863 

19.1572 

7.1596 

303 

91809 

27818127 

17.4069 

6.7166 

368 

1.35424 

49836032 

19.1833 

7.1661 

S04 

92416 

28094464 

17.4.356 

6.7240 

369 

136161 

50243409 

19.2094 

7.1726 

805 

93025 

28372625 

17.4642 

6.7313 

370 

136900 

50653000 

19.2.354 

7.1791 

306 

93636 

28652616 

17.4929 

6.7387 

371 

137641 

51064811 

19.2614 

7.1555 

307 

94249 

28934443 

17.5214 

6.7460 

372 

138384 

51478848 

19.2873 

7.1920 

306 

94864 

29218112 

17.5499 

6.75.33 

373 

139129 

51895117 

19.3132 

7.1984 

309 

95481 

29503629 

17.5784 

6.7606 

374 

139876 

52313624 

19.3391 

7.2048 

SIO 

96100 

29791000 

17.6068 

6.7679 

375 

140625 

52734375 

19.3649 

7.2112 

311 

96721 

30080231 

17.6352 

6.7752 

376 

141376 

53157376 

19.3907 

7.217T 

812 

97344 

30371328 

17.6635 

6.7824 

377 

142129 

53582633 

19.4165 

7.2240 

313 

97969 

30664297 

17.6918 

6.7897 

378 

142884 

54010152 

19.4422 

7.2304 

314 

98596 

30959144 

17.7200 

6.7969 

379 

143641 

54439939 

19.4679 

7.236« 

815 

99226 

31255875 

17.7482 

6.8041 

380 

144400 

54872000 

19.4936 

7.2433 

TABLE    XXIX. — SQUARES,    CUBES,    AND    ROOTS.       399 


TABIiE  of  Squares,  Cnbes,  Square  Roots,  and  Cube  Roots, 
of  Xambers  from  1  to  1000  — (Continued.) 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

No. 

Square. 

Cube. 

8q,  Rt. 

C.  Rt. 

381 

145161 

55306341 

19.5192 

7.2495 

446 

198916 

88716536 

21.1187 

7.6403 

382 

145924 

55742968 

19.5448 

7.2558 

447 

199809 

89314623 

21.1424 

7.6460 

383 

146689 

56181887 

19.5704 

7.2622 

448 

200704 

89915392 

21.1660 

7.6517 

384 

147456 

56623104 

,  19.5959 

7.2685 

449 

201601 

90518849 

21.1896 

7.6574 

385 

148225 

57066625 

19.6214 

7.2748 

450 

202500 

91125000 

21.2132 

7.6631 

386 

148996 

57512456 

19.6469 

7.2811 

451 

203401 

91733851 

21.2368 

7.6688 

387 

149769 

57960603 

19.6723 

7.2874 

452 

204304 

92345408 

21.2603 

7.6744 

388 

150544 

58411072 

19.6977 

7.2936 

453 

205209 

92959677 

21.28.38 

7.6801 

389 

151321 

58863869 

19.72.31 

7.2999 

454 

206116 

93576664 

21. .3073 

7.6857 

390 

152100 

59319000 

19.7484 

7.3061 

455 

207025 

94196375 

21.3307 

7.6914 

391 

152881 

59776471 

19.7737 

7.3124 

456 

207936 

94818816 

21. .3542 

7.6970 

392 

153664 

60236288 

19.7990 

7.3186 

457 

208849 

95443993 

21.3776 

7.7026 

393 

154449 

60698457 

19.8242 

7.3248 

458 

209764 

96071912 

21.4009 

7.7082 

394 

155236 

61162984 

19.8494 

7.3310 

459 

210681 

96702579 

21.4243 

7.7138 

395 

156025 

61629875 

19.8746 

7.3372 

460 

211600 

97336000 

21.4476 

7.7194 

396 

156816 

62099136 

19.8997 

7.3434 

461 

212521 

97972181 

21.4709 

7.7250 

397 

157609 

62570773 

19.9249 

7.3496 

462 

213444 

98611128 

21.4942 

7.7306 

398 

158404 

63044792 

19.9499 

7.. 3558 

463 

214369 

99252847 

21.5174 

7.7362 

399 

159201 

63521199 

19.9750 

7.3619 

464 

215296 

99897344 

21.5407 

7.7418 

400 

160000 

64000000 

20. 

7.3681 

465 

216225 

100544625 

21.5639 

7.7473 

401 

160801 

64481201 

20.0250 

7.3742 

466 

217156 

101194696 

21.5879 

7.7529 

402 

161604 

64964808 

20.0499 

7.3803 

467 

218089 

101847563 

21.6102 

7.7584 

403 

162409 

65450827 

20.0749 

7.3864 

468 

219024 

102503232 

21.6333 

7.7639 

404 

163216 

65939264 

20.0998 

7.3925 

469 

219961 

103161709 

21 .6564 

7.7695 

405 

164025 

66430125 

20.1246 

7.3986 

470 

220900 

103823000 

21.679ft 

7.7750 

406 

164836 

66923416 

20.1494 

7.4047 

471 

221841 

104487111 

21.7025 

7.7805 

407 

165649 

67419143 

20.1742 

7.4108 

472 

222784 

105154048 

21.7256 

7.7860 

408 

166464 

67917312 

20.1990 

7.4169 

473 

223729 

105823817 

21.7486 

7.7915 

409 

167281 

68417929 

20.2237 

7.4229 

474 

224676 

106496424 

21.7716 

7.7970 

410 

168100 

68921000 

20.2485 

7.4290 

475 

225625 

107171875 

21.7945 

7.8025 

411 

168921 

69426531 

20.2731 

7.4.350 

476 

226576 

107850176 

21.8174 

7.8079 

412 

169744 

69934528 

20.2978 

7.4410 

477 

227529 

108531333 

21.8403 

7.8134 

413 

170569 

70444997 

20.3224 

7.4470 

478 

228484 

109215352 

21.8632 

7.8188 

414 

171396 

70957944 

20.3470 

7.4530 

479 

229441 

109902239 

21.8861 

7.8243 

413 

172225 

71473375 

20.3715 

7.4590 

480 

230400 

110592000 

21.9089 

7.8297 

416 

173056 

71991296 

20.3961 

7.4650 

481 

231.361 

111284641 

21.9317 

7.8352 

417 

173889 

72511713 

20.4206 

7.4710 

482 

232324 

111980168 

21.9545 

7.8406 

418 

174724 

73034632 

20.4450 

7.4770 

483 

233289 

112678587 

21.9773 

7.8460 

419 

175561 

73560059 

20.4695 

7.4829 

484 

234256 

11.3379904 

22. 

7.8514 

420 

176400 

74088000 

20.4939 

7.4889 

485 

235225 

114084125 

22.0227 

7.8568 

421 

177241 

74618461 

20.5183 

7.4948 

486 

236196 

114791256 

22.0454 

7.8622 

422 

178084 

75151448 

20.5426 

7.5007 

487 

237169 

115501303 

22.0681 

7.8676 

423 

178929 

75686987 

20.5670 

7.5067 

488 

238144 

116214272 

22.0907 

7.87.30 

424 

179776 

76225024 

20.5913 

7.5126 

489 

239121 

116930169 

22.1133 

7.8784 

425 

,  180625 

76765625 

20.6155 

7.5185 

490 

240100 

117649000 

22.1359 

7.8837 

426 

181476 

77308776 

20.6398 

7.5244 

491 

241081 

118370771 

22.1585 

7.8891 

427 

182329 

77854483 

20.6640 

7.5302 

492 

242064 

119095488 

22.1811 

7.8944 

428 

183184 

78402752 

20.6882 

7.5361 

493 

243049 

119823157 

22.2036 

7.8998 

429 

184041 

78953589 

20.7123 

7.5420 

494 

244036 

120553784 

22.2261 

7.9051 

430 

184900 

79507000 

20.7364 

7.5478 

495 

245025 

121287375 

22.2486 

7.9105 

431 

185761 

80062991 

20.7605 

7.5537 

496 

246016 

122023936 

22.2711 

7.915a 

432 

186624 

80621568 

20.7846 

7.5595 

497 

247009 

122763473 

22.2935 

7.9211 

433 

187489 

81182737 

20.8087 

7.5654 

498 

248004 

123505992 

22.3159 

7.9264 

434 

188356 

81746504 

20.8327 

7.5712 

499 

249001 

124251499 

22.3383 

7.9317 

435 

189225 

82312875 

20.8567 

7.5770 

500 

250000 

12500000Q 

22.3607 

7.9370 

436 

190096 

82881856 

20.8806 

7.5828 

501 

251001 

125751501 

22.38.30 

7.942S 

437 

190969 

8345;H53 

20.9045 

7., 5886 

502 

252004 

126506008 

22.4054 

7.9476 

438 

191844 

84027672 

20.9284 

7.. 5944 

503 

253009 

12726.3527 

22.4277 

7.9523 

439 

192721 

84604519 

20.9523 

7.6001 

504 

254016 

128024064 

22.4499 

7.9581 

440 

193600 

85184000 

20.9762 

7.6059 

505 

255025 

128787625 

22.4722 

7.9634 

441 

194481 

85766121 

21. 

7.6117 

506 

256036 

129554216 

22.4944 

7.9686 

442 

195364 

86350888 

21.0238 

7.6174 

507 

257049 

130323843 

22.5167 

7.9739 

443 

196249 

86938307 

21.0476 

7.6232 

508 

258064 

131096512 

22.5389 

7.9791 

444 

197136 

87528384 

21.0713 

7.6289 

509 

259081 

131872229 

22.5610 

7. 9845 

44». 

198025 

8S12112d 

21.0959 

1Mi6 

510 

1«0100 

13a65".000 

22.58U 

i.9m% 

400      TABLE    XXIX. — SQUARES,   CUBES,   AND    ROOTS. 


TABIi£  of  Panares,  Cubes,  Sqnare  Roots,  and  Cnbe  Root», 
OJ^STninbers  from  1  to  1000  —  (Continued.) 


No. 

Square. 

Cube. 

Sq.  Kt. 

cut. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Kt. 

611 

261121 

133432831 

22.6053 

7.9948 

576 

331776 

191102976 

24. 

8.3203 

612 

262144 

134217728 

22.6274 

8. 

577 

332929 

192100033 

24.0208 

8.3251 

613 

263169 

135005697 

22.6495 

8.0052 

578 

334084 

193100552 

24.0416 

8.3300 

5U 

264196 

135796744 

22.6716 

8.0104 

579 

335241 

194104539 

24.0624 

8.3348 

515 

265225 

136590875 

22.6936 

8.0156 

580 

336400 

195112000 

24.0832 

«.3396 

516 

266256 

137.388096 

22.7156 

8.0208 

581 

337561 

196122941 

24.1039 

8.3443 

517 

267289 

138188413 

22.7376 

8.0260 

582 

338724 

197137368 

24.1247 

8.3491 

518 

268324 

138991832 

22.7596 

8.0311 

583 

339889 

198155287 

24.1454 

8.3539 

619 

269361 

139798359 

22.7816 

8.0363 

584 

341056 

199176704 

24.1661 

8.358T 

520 

270400 

140608000 

22.8035 

8.0415 

585 

342225 

200201625 

24.1868 

8.3634 

521 

271441 

141420761 

22.8254 

8.0466 

586 

343396 

201230056 

24.2074 

8.3682 

622 

272484 

142236648 

22.8473 

8.0517 

587 

344569 

202262003 

24.2281 

8.3730 

623 

273529 

143055667 

22.8692 

8.0569 

588 

345744 

203297472 

24.2487 

8.377T 

524 

274576 

143877824 

22.8910 

8.0620 

589 

346921 

204336469 

24.2693 

8.3825 

525 

275625 

144703125 

22.9129 

8.0671 

590 

348100 

205379000 

24.2899 

8.3872 

626 

276676 

145531576 

22.9347 

8.0723 

591 

349281 

206425071 

24.3105 

8.3919 

527 

277729 

146363183 

22.9565 

8.0774 

592 

350464 

207474688 

24.3311 

8.3967 

628 

278784 

147197952 

22.9783 

8.0825 

593 

351649 

208527857 

24.3516 

8.4014 

529 

279841 

148035889 

23. 

8.0876 

594 

352836 

209584584 

24.3721 

8.4061 

330 

280900 

148877000 

23.0217 

8.0927 

595 

354025 

210644875 

24.3926 

8.4108 

531 

281961 

149721291 

23.0434 

8.0978 

596 

355216 

211708736 

24.4131 

8.4155 

532 

283024 

150568768 

23.0651 

8.1028 

597 

356409 

212776173 

24.4336 

8.4202 

533 

284089 

151419437 

23.0868 

8.1079 

598 

357604 

213847192 

24.4540 

8.4249 

634 

285156 

152273304 

23.1084 

8.1130 

599 

358801 

214921799 

24.4745 

8.4296 

535 

286225 

153130375 

23.1301 

8.1180 

600 

360000 

216000000 

24.4949 

8.4343 

536 

287296 

153990656 

23.1517 

8.12.31 

601 

361201 

217081801 

24.5153 

8.4390 

537 

288369 

154854153 

23.1733 

8.1281 

602 

362404 

21S167208 

24.5357 

8.4437 

538 

289444 

155720872 

23.1948 

8.1332 

603 

363609 

219256227 

24.5561 

8.4484 

539 

290521 

156590819 

23.2164 

8.1.382 

604 

364816 

22034&864 

24.5764 

8.4530 

540 

291600 

157464000 

23.2379 

8.1433 

605 

366025 

221445125 

24.5967 

8.4577 

541 

292681 

158340421 

23.2594 

8.1483 

606 

367236 

222545016 

24.6171 

8.4623 

542 

293764 

159220088 

23.2809 

8.1533 

607 

368449 

223648543 

24.6374 

8.4670 

543 

294849 

160103007 

23.3024 

8.1583 

608 

369664 

224755712 

24.6577 

8.4716 

544 

295936 

160989184 

23.3238 

8.16:« 

609 

370881 

22586652S 

24.6779 

8.4763 

545 

297025 

161878625 

23.3452 

8.1683 

610 

372100 

226981000 

24.6982 

8.4809 

546 

298116 

162771336 

23.3666 

8.1733 

611 

373321 

228099131 

24.7184 

8.4856 

547 

299209 

163667323 

23.3880 

8.1783 

612 

374544 

229220928 

24.7386 

8.4902 

548 

300304 

164566592 

23.4094 

8.1833 

613 

375769 

230346397 

24.7588 

8.4948 

549 

301401 

165469149 

23.4307 

8.1882 

614 

376996 

231475544 

24.7790 

8.4994 

550 

302500 

166375000 

23.4521 

8.1932 

615 

378225 

232608375 

24.7992 

8.6040 

551 

303601 

167284151 

23.4734 

8.1982 

616 

379456 

233744896 

24.8193 

8.5086 

652 

304704 

168196608 

23.4947 

8.2031 

617 

380689 

234885113 

24.8395 

8.5132 

653 

.305809 

169112377 

23.5160 

8.2081 

618 

381924 

236029032 

24.8596 

8.5178 

554 

306916 

170031464 

23.5372 

8.2130 

619 

.383161 

237176659 

24.8797 

8.5224 

555 

308025 

170953875 

23.5584 

8.2180 

620 

384400 

238328000 

24.8998 

8.5270 

556 

309136 

171879616 

23.5797 

8.2229 

621 

385641 

239483061 

24.9199 

8.5316 

557 

310249 

172808693 

23.6008 

8.2278 

622 

386884 

240641848 

24.9399 

8.5.362 

558 

311364 

173741112 

23.6220 

8.2327 

623 

388129 

241804367 

24.9600 

8.5408 

559 

312481 

174676879 

23.6432 

8.2377 

624 

389376 

242970624 

24.9800 

8.5453 

560 

313600 

175616000 

23.6643 

8.2426 

625 

390625 

244140625 

25. 

8.5499 

561 

314721 

176558481 

23.6854 

8.2475 

626 

391876 

245314376 

25.0200 

8.5544 

562 

315844 

177504328 

23.7065 

8.2524 

627 

393129 

246491883 

25.0400 

8.5590 

563 

316969 

178453547 

23.7276 

8.2573 

628 

394384 

247673152 

25.0599 

8.5635 

564 

318096 

179406144 

23.7487 

8.2621 

629 

395641 

248858189 

25.0799 

8.5681 

565 

319225 

180362125 

23.7697 

8.2670 

630 

396900 

250047000 

25.0998 

8.5726 

666 

320356 

1813214% 

23.7908 

8.2719 

631 

398161 

251239591 

25.1197 

8.5772 

567 

321489 

182284263 

23.8118 

8.2768 

632 

399424 

252435968 

25.1396 

8.5817 

568 

322624 

183250432 

23.8328 

8.2816 

633 

400689 

253636137 

25.1595 

8.5862 

569 

323761 

184220009 

23.8537 

8.2865 

634 

401956 

2548*0104 

25.1794 

8.5907 

570 

324900 

185193000 

23.8747 

8.2913 

635 

403225 

256047875 

25.1992 

8.5952 

671 

326041 

186169411 

23.8956 

8.2962 

636 

404496 

257259456 

25.2190 

8.5997 

672 

327184 

187149248 

23.9165 

8.3010 

637 

405769 

258474853 

25.2389 

8.6043 

673 

328329 

188132517 

23.9374 

8.3059 

638 

407044 

259694072 

25.2587 

8.6088 

674 

329476 

189119224 

23.9583 

8.3107 

639 

408321 

260917119 

25.2784 

8.6133 

6TS 

330625 

190109375 

23.9792 

8.3155 

640 

409600 

262144000 

25.2982 

8.6177 

TABLE    XXIX. — SQUARES,    CUBES,    AND    ROOTS.       401 


TABIiE  of  Squares,  Cubes,  Sqnarc  Roots,  and  Tnbe  Roots, 
of  Xuinbers  from  1  to  1000  —  (Continued.) 


No. 

Square. 

Cube. 

Sq.  Kt. 

cut. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

641 

410881 

263374721 

25.3180 

8.6222 

706 

498436 

351895816 

26.5707 

8.9043 

642 

412164 

264609288 

25.3377 

8.6267 

707 

499849 

353393243 

26.5895 

8.9085 

643 

413449 

265847707 

25.3574 

8.6312 

708 

501264 

354894912 

26.6083 

8.9127 

644 

414736 

2C70899S4 

25.3772 

8.6357 

709 

502681 

356400829 

26.6271 

8.9169 

645 

416025 

268336125 

25.3969 

8.6401 

710 

504100 

357911000 

26.6458 

8.9211 

646 

417316 

269586136 

25.4165 

8.6446 

711 

505521 

359425431 

26.6646 

8.9253 

647 

418609 

270840023 

25.4362 

8.6490 

712 

506944 

360944128 

26.6833 

8.9295 

648 

419904 

272097792 

25.4558 

8.6535 

713 

508369 

362467097 

26.7021 

8.9337 

649 

421201 

273359449 

25.4755 

8.6579 

714 

509796 

363994344 

26.7208 

8.9378 

650 

422500 

274625000 

25.4951 

8.6624 

715 

511225 

365525875 

26.7395 

8.9420 

651 

423801 

275894451 

25.5147 

8.6668 

716 

512656 

367061696 

26.7582 

8.9462 

652 

425104 

277167808 

25.5343 

8.6713 

717 

514089 

368601813 

26.7769 

8.9503 

653 

426409 

278445077 

25.5539 

8.6757 

718 

515524 

370146232 

26.7955 

8.9545 

654 

427716 

279726264 

25.5734 

8.6801 

719 

516961 

371694959 

26.8142 

8.9587 

655 

429025 

281011375 

25.5930 

8.6845 

720 

518400 

373248000 

26.8328 

8.9628 

656 

430336 

282300416 

25.6125 

8.6890 

721 

519841 

374805361 

26.8514 

8.%70 

657 

431649 

283593393 

25.6320 

8.6934 

722 

521284 

376367048 

26.8701 

8.9711 

658 

432964 

284890312 

25.6515 

8.6978 

723 

522729 

377933067 

26.8887 

8.9752 

659 

434281 

286191179 

25.6710 

8.7022 

724 

524176 

379503424 

26.9072 

8.9794 

660 

435600 

287496000 

25.6905 

8.7066 

725 

525625 

381078125 

26.9258 

8.9835 

661 

436921 

288804781 

25.7099 

8.7110 

726 

527076 

382657176 

26.9444 

8.9876 

662 

438244 

J90117528 

25.7294 

8.7154 

727 

528529 

384240583 

26.9629 

8.9918 

663 

439569 

291434247 

25.7488 

8.7198 

728 

529984 

385828352 

26.9815 

8.9959 

664 

440896 

292754944 

25.7682 

8.7241 

729 

531441 

387420489 

27. 

9. 

665 

442225 

294079625 

25.7876 

8.7285 

730 

532900 

389017000 

27.0185 

9.0041 

666 

443556 

295408296 

25.8070 

8.7329 

731 

534361 

390617891 

27.0370 

9.008a 

667 

444889 

296740963 

25.8263 

8.7373 

732 

535824 

392223168 

27.0555 

9.0123 

668 

446224 

298077632 

25.8457 

8.7416 

733 

537289 

393832837 

27.0740 

9.0164 

669 

447661 

299418309 

25.8650 

8.7460 

734 

538756 

395446904 

27.0924 

9.0205 

670 

448900 

300763000 

25.8844 

8.7503 

735 

540225 

397065375 

27.1109 

Q.0246 

671 

450241 

302111711 

25.9037 

8.7547 

736 

541696 

398688256 

27.1293 

9.0287 

672 

451584 

303464448 

25.9230 

8.7590 

737 

543169 

400315553 

27.1477 

9.0328 

673 

452929 

304821217 

25.9422 

8.7634 

738 

544644 

401947272 

27.1662 

9.0369 

674 

454276 

306182024 

25.9615 

8.7677 

739 

546121 

403583419 

27.1816 

9.0410 

675 

455625 

307546875 

25.9808 

8.7721 

740 

547600 

405224000 

27.2029 

9.0450 

676 

456976 

308915776 

26. 

8.7764 

741 

549081 

406869021 

27.2213 

9.0491 

677 

458329 

310288733 

26.0192 

8.7807 

742 

550564 

408518488 

27.2397 

9.0532 

678 

459684 

311665752 

26.0384 

8.7850 

743 

552049 

410172407 

27.2580 

9.0572 

679 

461041 

313046839 

26.0576 

8.7893 

744 

553536 

411830784 

27.2764 

9.0613 

680 

462400 

314432000 

26.0768 

8.7937 

745 

555025 

413493625 

27.2947 

9.0654 

681 

463761 

315821241 

26.0960 

_  8.7980 

746 

556516 

415160936 

27.3130 

9.06S4 

682 

465124 

317214568 

26.1151 

8.8023 

747 

558009 

416832723 

27.3313 

9.07.S5 

683 

466489 

318611987 

26.1343 

8.8066 

748 

559504 

418508992 

27.3496 

9.0775 

684 

467856 

320013504 

26.1534 

8.8109 

749 

561001 

420189749 

27.3679 

9.0816 

685 

469225 

321419125 

26.1725 

8.8152 

750 

562500 

421875000 

27.3861 

9.0856 

686 

470596 

322828856 

26.1916 

8.8194 

751 

664001 

423564751 

27.4044 

9.0896 

687 

471969 

324242703 

26.2107 

8.8237 

752 

565504 

425259008 

27.4226 

9.0937 

688 

473344 

325660672 

26.2298 

8.8280 

753 

567009 

426957777 

27.4408 

9.0977 

689 

474721 

327082769 

26.2488 

8.8323 

754 

568516 

428661064 

27.4591 

9.1017 

C90 

476100 

328509000 

26.2679 

8.8366 

755 

570025 

430368875 

27.4773 

9.1057 

691 

477481 

329939371 

26.2869 

8.8408 

756 

571536 

432081216 

27.4955 

.1098 

692 

478864 

331373888 

26.3059 

8.8451 

757 

573049 

433798093 

27.5136 

9.11.38 

693 

480249 

332812557 

26.3249 

8.8493 

758 

574564 

435519512 

27.5318 

9.1178 

694 

481636 

334255384 

26.3439 

8.8536 

759 

576081 

437245479 

27.5500 

9.1218 

«95 

483025 

335702375 

26.3629 

8.8578 

760 

577600 

438976000 

27.5681 

9.1258 

6% 

484416 

337153536 

26.3818 

8.8621 

761 

579121 

440711081 

27.5862 

9.1298 

697 

485809 

33H608873 

26.4008 

8.8663 

762 

580644 

442450728 

27.6043 

9.1338 

698 

487204 

34006H392 

26.4197 

8.8706 

763 

582169 

444194947 

27.6225 

9.1378 

699 

488601 

341532099 

26.43f<6 

8.8748 

764 

583696 

445943744 

27.6405 

9.1418 

700 

490000 

343000000 

26.4575 

8.8790 

765 

585225 

447697125 

27.6586 

9.1458 

701 

491401 

344472101 

26.476! 

8.8833 

766 

5867.56 

449455096 

27.6767 

9.1498 

702 

492JS04 

34594840K 

26.4953 

H.HH75 

767 

588289 

451217663 

27.6948 

9.1537 

703 

494209 

3t742«»-.'7 

2<i.51ll 

8  8917 

768 

589824 

4529848;i2 

27.7128 

9.1577 

704 

495616 

348913664 

26.5330 

8.89.-.9 

769 

591361 

454756609 

27.7308 

W.1617 

105 

4U7025 

350402625 

26.5518 

3.9001 

770 

592900 

156533000 

27.7489 

9.1657 

402      TABLE    XXIX.— SQUARES.    CUBES,    AND    ROOTS. 


TABL.E  of  Squares,  Cubes,  Square  Hoots,  aiid  Cube  Roots, 
of  JK^umbers  from  1  to  1000  — (Continues.; 


No. 


771 
772 
773 
774 
775 

776 
777 

77S 
779 
780 

781 
782 
783 
784 
785 

786 
787 
788 
789 
790 

791 
792 
793 
794 
79i5 

796 
797 
798 
199 
POO 

J»l 

«02 
803 
804 
805 

806 
807 
808 
809 
810 

811 
812 
813 
814 
815 

816 
817 
818 
819 
820 

821 
822 
8-23 
824 
825 

826 

827 
828 
829 
830 

a31 
832 
833 
834 
835 


Square. 


594441 
595984 
597529 
599076 
600625 

602176 
603729 
605284 
606H41 
608400 

609961 
6U524 
613089 
614656 
616225 

617796 
619369 
620944 
622521 
624100 

625681 
627264 
628S49 
630436 
632026 

633616 
635209 
63&S04 
638101 
640000 

641601 
643204 
6H809 
646116 
648025 

649636 
651249 
652864 
654481 
656100 

657721 
659344 
660969 
662596 
664225 

665856 
667489 
669124 
670761 
672400 

674041 
675684 
677329 
678976 
680625 

682276 
683929 
685584 
687241 
688900 

690561 
692224 
693889 
695556 
697225 


Cube. 


45S314011 
460099648 
46ia89917 
463684824 
465484375 

467288576 
469097433 
470910952 
472729139 
474552000 

476379541 
478211768 
480048687 
481890304 
4837366-25 

485587656 
487443403 
489303872 
491169069 
493039000 

494913671 
496793088 
498677257 
500566184 
502459875 

504358336 
506261573 
508169592 
510082399 
512000000 

513922401 
515849608 
517781627 
519718464 
521660125 

523606616 
525557943 
527514112 
529475129 
531441000 

533411731 

535387328 
537367797 
539353144 
54134:1375 

543338496 
545338513 
547343432 
549353259 
551368000 

553387661 
555412248 
557441767 
559476224 
561515625 

563559976 
5656092h3 
567fi63552 
569722789 
571787000 

573856191 
575930368 
57S()09537 
580093704 
582182875 


Sq.  Bt. 


27.7669 
27.7S49 
27.8029 
27.8209 
27.8388 

27.8568 
27.8747 
27.8927 
27.9106 
27.9285 

27.9464 

27.9643 

27.9821 

28. 

28.0179 

28.0357 
28.0535 
28.0713 
28.0891 
28.1069 

28.1247 
28.1425 
28.1603 
28.1780 
28.1957 

28.2135 
28.2312 
28.2489 
28.2666 
28.2843 

28.3019 
28.3196 
28.3373 
28.3549 
28.3725 

28.. 3901 
28.4077 
28.4253 
28.4429 
28.4605 

28.4781 
28.4956 
28.5132 
28.5307 
28.5482 

28.5657 
28.5832 
28.6007 
28.6182 
28.6356 

28.6531 
2S.6705 
28.6880 
28.70.54 
28.7228 

28.7402 
28.7576 
28.7750 
28.7924 
28.8097 

28.8271 
28.8444 
28.8617 
28.8791 
38.8964 


C.  Kt. 


No. 


Square.     Cube,    j  Sq.  Rt. 


9.1696 
9.1736 
9.1775 
9.1815 
9.1855 

9.1894 
9.1933 
9.1973 
9.2012 
9.2052 

9.2091 
9.21.30 
9.2170 
9.2209 
9.2248 

9.2287 
9.2326 
9.2365 
9.2404 
9.2443 

9.2482 
9.2521 
9.2560 
9.2599 
9.2638 

9.2677 
9.27ie 
9.2754 
9.2793 
9.2832 

9.2870 
9.2909 
9.2948 
9.  •2986 
9.3025 

9..3063 
9.3102 
9.3140 
9.3179 
9.3217 

9.3255 
9.3294 
9.3332 
9..3370 


8:^6 
837 
a38 
839 
840 

841 

842 
843 
844 
845 

846 
847 
848 
849 
850 

851 
852 
853 
854 
855 

8,56 

857 
858 
8.J3 
860 

861 
862 
863 
864 
865 

866 

867 
868 
869 
670 

871 
872 
873 
874 
875 

876 
877 
878 
879 


9.3447 

881 

9.3485 

882 

9..3523 

883 

9.3561 

8.-S4 

9.3599 

885 

9..3637 

886 

9.3675 

R87 

9.3713 

888 

9.3751 

889 

9.3789 

890 

9..3«27 

891 

9.3865 

892 

9.3902 

893 

9.3940 

894 

9.3978 

895 

9.4016 

896 

9.4053 

897 

9.4091 

898 

9.4129 

899 

9.4166 

900  1 

698896 
700569 
702244 
703921 
705600 

707281 
708964 
710649 
712.3.36 
714025 

715716 
717409 
719104 
720801 
722500 

724201 
725904 
727609 
729316 
731025 

732736 
734449 
736164 
737881 
739600 

741321 
743044 
744769 
746496 
748225 

749956 
751689 
753424 
755161 
756900 

758641 
760384 
7621 29 
763876 
765625 

767376 
769129 
770884 
7726+1 
774400 

776161  - 

777924 

779689 

781456 

783225 

784996 
786769 
788544 
790321 
792100 

793881 
795664 
797449 
799-236 
801025 

802816 
804609 
806404 
808201 
810U00 


584277056 
58637S253 
588480472 
590589719 
592704000 

594823321 
596947688 
599077107 
601211584 
603351125 

6054957361 
607645423 
609800192 
611960049 
614125000 

616295051 
618470208 
620650477 
622835864 
625026375 

657222016 
629422793 
631628712 
633839779 
636056000 

6.38277381 
640503928 
642735647 
644972544 
647214625 

649461896 
651714363 
653972032 
656234909 
658503000 

660776311 
663054848 
665338617 
667627624 
669921875 

672221376 
674526133 
676836152 
679151439 
681472000 

683797841 
686128968 
688465387 
690807104 
693154125 

695506456! 
697^64103 
700227072 
702595369 
704969000 

707347971 
709732288 
712121957 
714516984 
716917375 

719323136 
721734273 
724150792 
726572699 
729000000 


28.9137 
28.9310 
28.9482 
28.9655 
28.9828 

29. 

29.0172 
29.0345 
29.0517 
29.0689 

J9.0861 
29.1033 
29.1204 
29.1376 
29.1548 

29.1719 
29  1890 
29.2062 
29.2233 
29.2404 

29.2575 
29.2746 
29.2916 
29.3087 
29.3258 

29.3428 
29.3598 
29.3769 
29.3939 
29.4109 

29.4279 
29.4449 
29.4618 
29.4788 
29.4958 

29..5127 
29.5'2&6 
29.5466 
29.56;{5 
29.5804 

29.5973 
29.6142 
29.6311 
29.6479 
29.6648 

29.6«16 
29.69h5 
29.7153 
29.7321 
29.7489 

29.7658 
29.7825 
29.7993 
29.8161 
29.8329 

29.8496 
29.8664 
29.8831 
29.8998 
29.9166 

29.9333 
29.9500 
29.9666 
29.9833 
30. 


C.  Kt. 


9.4204 
9.4241 
9.4279 
9.4316 
9.4;j54 

9.4.391 
9.4429 
9.4466 
9.4.503 
9.4541 

9.4,578 
9.4615 
9.4652 
9.4690 
9.4727 

9.4761 
9.4801 
9.4838 
9.4875 

9.491  :j 

9.4949 

9.498JJ 
9..5023 
9..5060 
9.5097 

9..5134 
9.5171 
9.5207 
9.5244 
9.523^ 

9..531  J 
9.5354 
9.5391 
9.5427 
9.5461 

9..5.501 
9.5.537 
9.,5574 
9..5610 
9.5647 

9.5683 
9.5719 
9.5756 
9.5792 
9.5828 

9.. 5865 
9.5901 
9.5937 
9.5973 
9.6010 

9.6046 
9.6082 
9.6118 
9.61,54 
9.6190 

9.6226 
9.6262 
9.6298 
9.6334 
9.6370 

9.6406 
9.6442 
9.6477 
9.6513 
9.6549 


TAHLE    XXIX. — SQUARES,    CUBES,    AND    HOOTS       Wd 

TABL.E  of  Squares,  dibos,  Square  Roots,  au<l  €nl>e  Roots, 
of  3f  uiubers  irom  1  to  1000  — (^Cominukd.) 


No. 


Square. 


901 
90-2 
903 

905 

906 
907 
908 
909 
910 

911 
312 
913 
914 
915 

910 
917 
918 
919 
920 

921 
922 
923 
921 
925 

926 

927 
928 
929 
930 

931 
932 
933 
931 
935 

936 
93- 
938 
939 
910 

911 
912 
913 
914 
945 

916 
917 
918 
9*9 
950 


811801 
813604 
815109 
817216 
819025 

820836 
822619 
821164 
826281 
828100 

829921 
831714 
833569 
835396 
837225 

839056 
810H89 
812721 
811561 
816100 

818211 
850084 
851929 
853776 
855625 

857176 
859329 
861181 
863011 
861900 

866761 
868621 
870189 
872356 
871225 

876096 
877969 
879^11 
881721 
883600 

885181 
887361 
889219 
891 136 
893025 

891916 
896S09 
898701 
900601 
902500 


Cube. 


731432701 
733870808 
736314327 
738763264 
741217625 

743677416 
746142643 
748613312 
751089129 
753571000 

756058031 
758550528 
761048497 
763551944 
766060875 

768575296 
771095213 
773620632 
776151559 

778688000 

781229961 
783777448 
786330167 
788889024 
791453125 

794022776 
796597983 
799178752 
801765089 
804357000 

806954491 
809557568 
812166237 
-14780501 
817400375 

820025856 
822656!)53 
825293672 
827936019 
830584000 

833237621 

835S968.Si8 
83S501hO7 
8U23238J 
843908C25 

846590536 
81927.H123 
851971392 
8546703 19 
857375000 


Sq.  Rt. 


30.0167 
30.0333 
30.0500 
30.0666 
30.0832 

30.0998 
30.1164 
30.1330 
30.1496 
30.1662 

30.1828 
30.1993 
30.2159 
.30.2324 
30.2490 

30.2655 
30.2820 
30.2985 
.30.3150 
30.3315 

.30.3180 
30.3645 
30.3809 
30.3974 
30.4138 

.30.4302 
30.4467 
30.4631 
30.4795 
30.4959 

30.5123 
30.5287 
30.5450 
.30.5614 
30.5778 

30.5941 
30.6105 
30.6268 
30.6131 
30.6594 

30.6757 
30.6920 
30.7083 
30.7246 
30.7409 

30.7571 
30.7734 
30.7896 
30.8058 
30.8221 


C.  Rt. 


No. 


9.6585 
9.6620 
9.6656 
9.6092 
9.6727 

9.6763 
9.6799 
9.6834 
9.6870 
9.6905 

9.6941 
9.6976 
9.7012 
9.7047 
9.7082 

9.7118 
9.7153 
9.7188 
9.7224 
9.7259 

9.7294 
9.7329 
9.7364 
9.7400 
9.7435 

9.7470 
9.7505 
9.7510 
9.7575 
9.7610 

9.7615 
9.7680 
9.7715 
9.7750 
9.7785 

9.7819 
9.7854 
9.7889 
9.7921 
9.7959 

9.7993 
9.8028 
9.80G3 
9.8097 
9.8132 

9.8167 
9.8201 
9.8236 
9.8270 
9.8305 


951 
952 
953 
954 
955 

956 
957 
958 
959 
960 

961 
962 
9G3 
964 
965 

966 
967 
968 
969 
970 

971 
972 
973 
974 
975 

976 
977 
978 
979 
980 

981 
982 
983 
984 
985 

986 
987 
988 
989 
990 

991 
992 
993 
994 
995 

996 
997 
998 
999 
1000 


Square. 


901401 
906304 
908209 
910116 
912025 

913936 
915849 
917764 
919681 
921600 

923521 
925444 
927369 
929296 
931225 

9331.56 
935089 
937024 
938961 
940900 

942S41 
911781 
916729 
918676 
950625 

952576 

951529 
956184 
958141 
960400 

962361 
961321 
966289 
968256  , 
970225 

972196 
971169 
976144 
978121 
980100 

982081 
981064 
986049 
988036 
990025 

992016 
994009 
996001 
998001 
1000000 


Cube. 


860085351 
862401408 
865523177 
868250664 
870983875 

873722816 
876467493 
879217912 
881974079 
884736000 

887503681 
890277128 
893056347 
895841344 
898632125 

901428696 
904231063 
907039232 
909853209 
912673000 

915498611 

918330048 

92116731 

924010424 

926859375 

929714176 
932574833 
935441352 
938313739 
941192000 

944076141 
946966168 
949862087 
952763904 
955671625 

958585256 
961504803 
9644.30272 
967361669 
970299000 

973242271 
976191488 
979146657 
982107784 
985074875 

988047936 
991026973 
991011992 
997002999 
1000000000 


Sq.  Rt. 


30.8383 
30.8545 
30.8707 
30.8869 
30.9031 

30.9192 
30.9354 
80.9516 
30.9677 
30.9839 

31. 

31.0161 

31.0322 

31.0483 

31.0644 

31.0805 
31.0966 
31.1127 
31.1288 
31.1448 

31.1609 
31.1769 
31.1929 
31.2090 
31.2250 

31.2410 
31.2570 
31.2730 
31.2890 
31.3050 

31.3209 
31.3369 
31.3528 
31.3688 
31.3847 

31.4006 
31.4166 
31.4325 
31.4484 
31.4643 

31.4802 
31.4960 
31.5119 
31.5278 
31.5436 

,31.5595 
31.5753 
31.5911 
31.6070 
31.6228 


C.  Rt. 


9.8.339 
9.8374 
9.8408 
9.8tl3 
9.8477 

9.8511 
9.8546 
9.8580 
9.8614 
9.8618 

9.8683 
9.8717 
9.8751 
9.8785 
9.8819 

9.8854 
9.8888 
9.8922 
9.8956 
9.8990 

9.9024 
9.9058 
9.909-2 
9.9126 
9.9160 

9.9194 

9.9227 
9.9261 
9.9295 
9.9329 

9.9.363 
9.9396 
9.9430 
9.9164 
9.9497 

9.9531 
9.9565 
9.9598 
9.9632 
9.9606 

9.9699 
9.9733 
9.9766 
9.9800 
9.9833 

9.9866 
9.9900 
9.99.33 
9.9967 
10. 


lo  find  the  square  or  ceibe  of  any  nliole  number  ending 
wilBi  ciphers.     Fir.st,  omit  all  the  fiual  tiphcr.s.    Take  from  the  table  the 

.square  or  cufic  (as  the  case  may  be)  of  the  rest  of  the  number.  To  this  square  add  ttvice  as  many 
ciphers  as  there  were  tiuul  ci|ihers  in  the  orignial  number.  To  the  ciihc  aiid  tliree  times  .isfinany  :i3 
in  the  original  number.  Thus,  for  905(Kr^  ;  90o2  — ,sl9025.  Add  twice  2  ci|ihers.  obtaining  8ii(025i;UOO. 
For  905003,  9053  =  711217625.     Add  3  times  2  ciphers,  obtaining  7412176250ii(MlOO. 


e,-^/eV'  o/^  ^^^'^y 


^.X,-^ 


A:- 


c:^,  -^i^ 


y...    „v-x- 


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